arxiv:2007.11769v3 [math.co] 25 feb 2021

Graphs isomorphisms under edge-replacements

and the family of amoebas

February 26, 2021

Yair Caro

Dept. of MathematicsUniversity of Haifa-Oranim

Tivon 36006, Israel

[email protected]

Adriana Hansberg

Instituto de MatematicasUNAM JuriquillaQueretaro, Mexico

[email protected]

Amanda Montejano

UMDI, Facultad de CienciasUNAM JuriquillaQueretaro, Mexico

[email protected]

Abstract

This paper offers a systematic study of a family of graphs called amoebas. Amoebasrecently emerged from the study of forced patterns in 2-colorings of EpKnq in thecontext of Ramsey-Turan theory and played an important role in extremal zero-sumproblems. Amoebas are graphs with a unique behavior with regards to the followingoperation: Let G be a graph of order n and let e P EpGq and e1 P EpGq Y teu. If thegraph G´ e` e1 is isomorphic to G, we say that eÑ e1 is a feasible edge-replacement.We call G a local amoeba if, for any two copies G1, G2 of G that are embedded in Kn,G1 can be transformed into G2 by a chain of feasible edge-replacements. On the otherhand, G is called global amoeba if there is an integer T ě 0 such that G Y tK1 is alocal amoeba for all t ě T . To model the dynamics of the feasible edge-replacementsof G, we define a group SG that satisfies that G is a local amoeba if and only ifSG – Sn. Via this algebraic setting, a deeper understanding of the structure of localamoebas and their intrinsic properties comes into light. Moreover, we present differentconstructions that prove the richness of these graphs families showing, among otherthings, that any connected graph can be a connected component of a global amoeba,that global amoebas can be very dense and that they can have, in proportion to theirorder, large clique and chromatic numbers. Also, a family of global amoeba trees witha Fibonacci-like structure and with arbitrary large maximum degree is constructed.

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1 Introduction

Graphs called amoebas first appeared in [10] where certain Ramsey-Turan extremal prob-lems were considered, which dealt with the existence of a given graph with a prescribedcolor pattern in 2-edge-colorings of the complete graph. More precisely, amoebas arosefrom the search of a graph family with certain interpolation properties that are suitablefor the techniques to show balanceability or omnitonal properties [10] (see also [9]). Forthe interested reader, we refer to [6, 15, 16, 23, 26, 27, 34] for more literature related tointerpolation techniques in graphs.

The feature that makes amoebas work are one-by-one replacements of edges, where,at each step, some edge is substituted by another such that an isomorphic copy of thegraph is created. We call such edge substitutions feasible edge-replacements. Similar edge-operations have been studied, for instance, in [11, 13, 14, 20, 27, 29]. As introduced in [9],a family F of graphs, all of them having the same number of edges, is called closed in agraph H if, for every two copies F, F 1 of members of F contained in H, there is a chainof graphs H1, H2, . . . ,Hk in H such that F “ H1, F 1 “ Hk, and, for 2 ď i ď k, Hi is amember of F and Hi obtained from Hi´1 by a feasible edge-replacement. Perhaps the mostwell-known closed family is the family of all spanning trees of a connected graph H, andthe edge-replacement operation given above is in fact the basic operation in the exchangeof bases in the cycle matroid MpHq of H. A graph G is a global amoeba precisely whentGu is a closed family in Kn (for n large enough), and it is a local amoeba if tGu is a closedfamily in KnpGq. Exactly this amoeba-property is the key-role to the usefulness of amoebasin interpolation theorems in Graph Theory and in zero-sum extremal problems [9], and inproblems about forced patterns in 2-colorings of the edges of Kn [10]. We note at this pointthat the amoebas defined in [10] correspond to the class of global amoebas.

For example, it can be easily checked that the path Pk on k ě 2 vertices is a globalamoeba: say P “ v1v2 . . . vk is embedded in Kn. Then we can remove the edge vk´1vk fromP and include the edge v1vk, so that the new graph is again a path on k vertices. Similarly,we can take any vertex v P V pKnqzV pP q and substitute the edge vk´1vk with the edgevk´1v. With these two operation-types, we can obtain a series of paths whose last memberis certain given copy P 1 of P . Clearly, such a chain of operations can happen if n is largeenough and it is not clear at a first look how large the n needs to be at least. Interestingly,it turns out that we just need n to have one unit larger than the order of P , and that occursfor any global amoeba. This is shown in Theorem 3.9, which is a major achievement ofthis paper. It is also easy to see that paths are local amoebas, too. However, for example,a complete graph minus an edge is a local amoeba but not a global amoeba (assuming wehave at least 4 vertices). We will give these and other more detailed examples further on,which will also be shown formally to which family they do or do not belong. The directconnection between these two graph families can already be seen in their formal definitionthat will be given further on (Definition 3.1).

A first encounter with amoebas gives the impression that such graphs are very rare andhave a very simple structure. This, however, is not the case and amoebas may have quitea complicated structure. Indeed, we will consider here different constructions with whichwe will show that any connected graph can be a connected component of a global amoeba,that global amoebas can be very dense (in fact, with as many as n2{4 edges, being n the

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order of the graph) and, that they can have very large chromatic number and cliques, too(as large as roughly half the order of the graph). Also, we introduce an interesting familyof global amoeba trees with a Fibonacci-like structure and with arbitrary large maximumdegree.

Most concepts and definitions concerning amoebas can be stated in graph theoreticallanguage. However, a group theoretical setting with which the dynamics of the edge re-placements are modeled – and which involve graph isomorphisms – will be necessary forproving several results. In particular, the proof of Theorem 3.8, which is a key elementfor many other results, employs group theoretical tools that via combinatorial language areunimaginable. This is the reason why we will first develop the algebraic theoretical settingso that we can then formalize all concepts and definitions by means of their group actionfeatures, and we will proceed further on with the theory this way. The research on amoebasis related to problems like switching in graphs [5, 18] and, with respect to aspects of groupaction language, to reconstruction problems in graphs [12, 17, 22]. Similar approaches thatdeal with graph isomorphisms can be found in [1, 28, 30]. For group theoretical concepts,we refer to [19].

The paper is organized as follows. In Section 2, we present the group theoretical back-ground with which we will be able to model how the operations that we will be performingon a graph G, the so-called feasible edge-replacements, work such that always isomorphiccopies of G are obtained. By means of this algebraic setting, we will formally introduce,in Section 3, the concepts of global amoeba and local amoeba (Definition 3.1), and wewill demonstrate several structural properties of both graph families, establishing also veryclearly the relation between them as well as their differences. In order to show the purposeof the results, we will illustrate with abundant examples. In Section 4, we will present someinteresting constructions of both local and global amoebas that will prove the richness ofthis family of graphs. In section 5 we exhibit extremal global amoebas with respect tosize, chromatic number and clique number. In Section 6, we will show how the developedtheory can be applied to all different kinds of examples, in fact those that were alreadymentioned (without proofs) in Section 3. In the final section, we provide the reader withseveral open problems which could bring more light to understanding this very interestingfamily of graphs called amoebas.

2 Theoretical setting

For integers m and n with m ă n we use the standard notation rns “ t1, 2, ..., nu andrm,ns “ tm,m ` 1,m ` 2, , ..., nu. Let Sn be the symmetric group, whose elements arepermutations of rns. The group of automorphisms of a graph G is denoted by AutpGq.Thus, AutpKnq – Sn where Kn is the complete graph of order n and, for any graph G oforder n, AutpGq – S for some S ď Sn. Let V “ tv1, v2, . . . , vnu be the set of vertices ofKn. Let G be a spanning subgraph of Kn defined by its edge set EpGq Ď EpKnq and letLG “ tij | vivj P EpGqu, where we do not distinguish between ij and ji. For each σ P Sn,we define λσ : V Ñ rns as the labeling of the vertices of Kn defined by λσpviq “ σpiq andconsider the copy Gσ of G embedded in Kn defined by

EpGσq “ tvσ´1piqvσ´1pjq | ij P LGu.

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Hence, each labeled copy of G embedded in Kn correspond to a permutation σ P Sn and viseversa. Observe that, for every (non-labeled) subgraph G1 of Kn isomorphic to G there are|AutpGq| different labelings of V that correspond to G1, that is, the set tσ P Sn |Gσ “ G1uhas |AutpGq| elements. Moreover, tσ P Sn |Gσ “ Gu – AutpGq. We will set

AG “ tσ P Sn |Gσ “ Gu.

Example 2.1. Let G “ P4 with V pP4q “ tv1, v2, v3, v4u and EpP4q “ tv1v2, v2v3, v3v4u.Thus, LG “ t12, 23, 34u and tσ P S4 |Gσ “ Gu “ tid, p14qp23qu – AutpP4q. For G1, theisomorphic copy of G defined by EpG1q “ tv1v3, v2v3, v2v4u, we have two permutations,namely p23q and p14q, that satisfy Gp23q “ Gp14q “ G1. See Figure 1 to visualize thecorresponding labelings and observe that, in all cases, EpGσq “ tvσ´1piqvσ´1pjq | ij P LGu.For example, if σ “ p23q then EpGσq “ tvσ´1piqvσ´1pjq | ij P LGu “ tv1v3, v3v2, v2v4u.

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1

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2v2

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42

3

G = G(14)(23)<latexit sha1_base64="NSlMDFE3abJAVjowzvChk6KM8ZA=">AAAB83icbZDLTsJAFIZPvSLeqi7dNIIJbEiLJLoxIXGBS0zkkkDTTIcDTJheMjMlIQ1Poiuj7nwTX8C3ccAuFPxX35z/n+Sc3485k8q2v4yNza3tnd3cXn7/4PDo2Dw5bcsoERRbNOKR6PpEImchthRTHLuxQBL4HDv+5G7hd6YoJIvCRzWL0Q3IKGRDRonSI880i43bhpeWnFq5VL0qz4ueWbAr9lLWOjgZFCBT0zM/+4OIJgGGinIiZc+xY+WmRChGOc7z/URiTOiEjLCnMSQBSjddbj63LoeRsNQYreX7dzYlgZSzwNeZgKixXPUWw/+8XqKGN27KwjhRGFId0d4w4ZaKrEUB1oAJpIrPNBAqmN7SomMiCFW6prw+31k9dh3a1YpjV5yHaqFey4rIwTlcQAkcuIY63EMTWkBhCs/wBu9GYjwZL8brT3TDyP6cwR8ZH9+bpY6m</latexit><latexit sha1_base64="NSlMDFE3abJAVjowzvChk6KM8ZA=">AAAB83icbZDLTsJAFIZPvSLeqi7dNIIJbEiLJLoxIXGBS0zkkkDTTIcDTJheMjMlIQ1Poiuj7nwTX8C3ccAuFPxX35z/n+Sc3485k8q2v4yNza3tnd3cXn7/4PDo2Dw5bcsoERRbNOKR6PpEImchthRTHLuxQBL4HDv+5G7hd6YoJIvCRzWL0Q3IKGRDRonSI880i43bhpeWnFq5VL0qz4ueWbAr9lLWOjgZFCBT0zM/+4OIJgGGinIiZc+xY+WmRChGOc7z/URiTOiEjLCnMSQBSjddbj63LoeRsNQYreX7dzYlgZSzwNeZgKixXPUWw/+8XqKGN27KwjhRGFId0d4w4ZaKrEUB1oAJpIrPNBAqmN7SomMiCFW6prw+31k9dh3a1YpjV5yHaqFey4rIwTlcQAkcuIY63EMTWkBhCs/wBu9GYjwZL8brT3TDyP6cwR8ZH9+bpY6m</latexit><latexit sha1_base64="NSlMDFE3abJAVjowzvChk6KM8ZA=">AAAB83icbZDLTsJAFIZPvSLeqi7dNIIJbEiLJLoxIXGBS0zkkkDTTIcDTJheMjMlIQ1Poiuj7nwTX8C3ccAuFPxX35z/n+Sc3485k8q2v4yNza3tnd3cXn7/4PDo2Dw5bcsoERRbNOKR6PpEImchthRTHLuxQBL4HDv+5G7hd6YoJIvCRzWL0Q3IKGRDRonSI880i43bhpeWnFq5VL0qz4ueWbAr9lLWOjgZFCBT0zM/+4OIJgGGinIiZc+xY+WmRChGOc7z/URiTOiEjLCnMSQBSjddbj63LoeRsNQYreX7dzYlgZSzwNeZgKixXPUWw/+8XqKGN27KwjhRGFId0d4w4ZaKrEUB1oAJpIrPNBAqmN7SomMiCFW6prw+31k9dh3a1YpjV5yHaqFey4rIwTlcQAkcuIY63EMTWkBhCs/wBu9GYjwZL8brT3TDyP6cwR8ZH9+bpY6m</latexit><latexit sha1_base64="NSlMDFE3abJAVjowzvChk6KM8ZA=">AAAB83icbZDLTsJAFIZPvSLeqi7dNIIJbEiLJLoxIXGBS0zkkkDTTIcDTJheMjMlIQ1Poiuj7nwTX8C3ccAuFPxX35z/n+Sc3485k8q2v4yNza3tnd3cXn7/4PDo2Dw5bcsoERRbNOKR6PpEImchthRTHLuxQBL4HDv+5G7hd6YoJIvCRzWL0Q3IKGRDRonSI880i43bhpeWnFq5VL0qz4ueWbAr9lLWOjgZFCBT0zM/+4OIJgGGinIiZc+xY+WmRChGOc7z/URiTOiEjLCnMSQBSjddbj63LoeRsNQYreX7dzYlgZSzwNeZgKixXPUWw/+8XqKGN27KwjhRGFId0d4w4ZaKrEUB1oAJpIrPNBAqmN7SomMiCFW6prw+31k9dh3a1YpjV5yHaqFey4rIwTlcQAkcuIY63EMTWkBhCs/wBu9GYjwZL8brT3TDyP6cwR8ZH9+bpY6m</latexit>





4

13

2

G = Gid<latexit sha1_base64="Z+xZM3ssbbTTxgjjoPSQgx04+10=">AAAB63icbZDNTgIxFIXv4B/iH+rSTSOYuCIzxEQ3JiQucImJ/ESYkE65QEM7M2k7JmTCU+jKqDvfxhfwbSw4CwXP6us9p8k9N4gF18Z1v5zc2vrG5lZ+u7Czu7d/UDw8aukoUQybLBKR6gRUo+AhNg03AjuxQioDge1gcjP324+oNI/CezON0Zd0FPIhZ9TY0UO5fl3vp3wwK/eLJbfiLkRWwcugBJka/eJnbxCxRGJomKBadz03Nn5KleFM4KzQSzTGlE3oCLsWQypR++li4xk5G0aKmDGSxft3NqVS66kMbEZSM9bL3nz4n9dNzPDKT3kYJwZDZiPWGyaCmIjMi5MBV8iMmFqgTHG7JWFjqigz9jwFW99bLrsKrWrFcyveXbVUu8gOkYcTOIVz8OASanALDWgCgxCe4Q3eHek8OS/O608052R/juGPnI9vSO2Nmg==</latexit><latexit sha1_base64="Z+xZM3ssbbTTxgjjoPSQgx04+10=">AAAB63icbZDNTgIxFIXv4B/iH+rSTSOYuCIzxEQ3JiQucImJ/ESYkE65QEM7M2k7JmTCU+jKqDvfxhfwbSw4CwXP6us9p8k9N4gF18Z1v5zc2vrG5lZ+u7Czu7d/UDw8aukoUQybLBKR6gRUo+AhNg03AjuxQioDge1gcjP324+oNI/CezON0Zd0FPIhZ9TY0UO5fl3vp3wwK/eLJbfiLkRWwcugBJka/eJnbxCxRGJomKBadz03Nn5KleFM4KzQSzTGlE3oCLsWQypR++li4xk5G0aKmDGSxft3NqVS66kMbEZSM9bL3nz4n9dNzPDKT3kYJwZDZiPWGyaCmIjMi5MBV8iMmFqgTHG7JWFjqigz9jwFW99bLrsKrWrFcyveXbVUu8gOkYcTOIVz8OASanALDWgCgxCe4Q3eHek8OS/O608052R/juGPnI9vSO2Nmg==</latexit><latexit sha1_base64="Z+xZM3ssbbTTxgjjoPSQgx04+10=">AAAB63icbZDNTgIxFIXv4B/iH+rSTSOYuCIzxEQ3JiQucImJ/ESYkE65QEM7M2k7JmTCU+jKqDvfxhfwbSw4CwXP6us9p8k9N4gF18Z1v5zc2vrG5lZ+u7Czu7d/UDw8aukoUQybLBKR6gRUo+AhNg03AjuxQioDge1gcjP324+oNI/CezON0Zd0FPIhZ9TY0UO5fl3vp3wwK/eLJbfiLkRWwcugBJka/eJnbxCxRGJomKBadz03Nn5KleFM4KzQSzTGlE3oCLsWQypR++li4xk5G0aKmDGSxft3NqVS66kMbEZSM9bL3nz4n9dNzPDKT3kYJwZDZiPWGyaCmIjMi5MBV8iMmFqgTHG7JWFjqigz9jwFW99bLrsKrWrFcyveXbVUu8gOkYcTOIVz8OASanALDWgCgxCe4Q3eHek8OS/O608052R/juGPnI9vSO2Nmg==</latexit><latexit sha1_base64="Z+xZM3ssbbTTxgjjoPSQgx04+10=">AAAB63icbZDNTgIxFIXv4B/iH+rSTSOYuCIzxEQ3JiQucImJ/ESYkE65QEM7M2k7JmTCU+jKqDvfxhfwbSw4CwXP6us9p8k9N4gF18Z1v5zc2vrG5lZ+u7Czu7d/UDw8aukoUQybLBKR6gRUo+AhNg03AjuxQioDge1gcjP324+oNI/CezON0Zd0FPIhZ9TY0UO5fl3vp3wwK/eLJbfiLkRWwcugBJka/eJnbxCxRGJomKBadz03Nn5KleFM4KzQSzTGlE3oCLsWQypR++li4xk5G0aKmDGSxft3NqVS66kMbEZSM9bL3nz4n9dNzPDKT3kYJwZDZiPWGyaCmIjMi5MBV8iMmFqgTHG7JWFjqigz9jwFW99bLrsKrWrFcyveXbVUu8gOkYcTOIVz8OASanALDWgCgxCe4Q3eHek8OS/O608052R/juGPnI9vSO2Nmg==</latexit> G0 = G(14)

<latexit sha1_base64="f0k3YIjcLNgdj2J8TMB2sOtDk08=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZQqIbExIXuMRELglMSKccoKFzse2YkAmvoSuj7nwYX8C3seAsFPxXX8//Nzn/8SLBlbbtLyuztr6xuZXdzu3s7u0f5A+PWiqMJcMmC0UoOx5VKHiATc21wE4kkfqewLY3uZn77UeUiofBvZ5G6Pp0FPAhZ1SbkVusn1/X+0nJqV7Miv18wS7bC5FVcFIoQKpGP//ZG4Qs9jHQTFCluo4daTehUnMmcJbrxQojyiZ0hF2DAfVRucli6Rk5G4aS6DGSxft3NqG+UlPfMxmf6rFa9ubD/7xurIdXbsKDKNYYMBMx3jAWRIdk3p0MuESmxdQAZZKbLQkbU0mZNhfKmfrOctlVaFXKjl127iqFWjU9RBZO4BRK4MAl1OAWGtAEBg/wDG/wbkXWk/Vivf5EM1b65xj+yPr4BtWdjcg=</latexit><latexit sha1_base64="f0k3YIjcLNgdj2J8TMB2sOtDk08=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZQqIbExIXuMRELglMSKccoKFzse2YkAmvoSuj7nwYX8C3seAsFPxXX8//Nzn/8SLBlbbtLyuztr6xuZXdzu3s7u0f5A+PWiqMJcMmC0UoOx5VKHiATc21wE4kkfqewLY3uZn77UeUiofBvZ5G6Pp0FPAhZ1SbkVusn1/X+0nJqV7Miv18wS7bC5FVcFIoQKpGP//ZG4Qs9jHQTFCluo4daTehUnMmcJbrxQojyiZ0hF2DAfVRucli6Rk5G4aS6DGSxft3NqG+UlPfMxmf6rFa9ubD/7xurIdXbsKDKNYYMBMx3jAWRIdk3p0MuESmxdQAZZKbLQkbU0mZNhfKmfrOctlVaFXKjl127iqFWjU9RBZO4BRK4MAl1OAWGtAEBg/wDG/wbkXWk/Vivf5EM1b65xj+yPr4BtWdjcg=</latexit><latexit sha1_base64="f0k3YIjcLNgdj2J8TMB2sOtDk08=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZQqIbExIXuMRELglMSKccoKFzse2YkAmvoSuj7nwYX8C3seAsFPxXX8//Nzn/8SLBlbbtLyuztr6xuZXdzu3s7u0f5A+PWiqMJcMmC0UoOx5VKHiATc21wE4kkfqewLY3uZn77UeUiofBvZ5G6Pp0FPAhZ1SbkVusn1/X+0nJqV7Miv18wS7bC5FVcFIoQKpGP//ZG4Qs9jHQTFCluo4daTehUnMmcJbrxQojyiZ0hF2DAfVRucli6Rk5G4aS6DGSxft3NqG+UlPfMxmf6rFa9ubD/7xurIdXbsKDKNYYMBMx3jAWRIdk3p0MuESmxdQAZZKbLQkbU0mZNhfKmfrOctlVaFXKjl127iqFWjU9RBZO4BRK4MAl1OAWGtAEBg/wDG/wbkXWk/Vivf5EM1b65xj+yPr4BtWdjcg=</latexit><latexit sha1_base64="f0k3YIjcLNgdj2J8TMB2sOtDk08=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZQqIbExIXuMRELglMSKccoKFzse2YkAmvoSuj7nwYX8C3seAsFPxXX8//Nzn/8SLBlbbtLyuztr6xuZXdzu3s7u0f5A+PWiqMJcMmC0UoOx5VKHiATc21wE4kkfqewLY3uZn77UeUiofBvZ5G6Pp0FPAhZ1SbkVusn1/X+0nJqV7Miv18wS7bC5FVcFIoQKpGP//ZG4Qs9jHQTFCluo4daTehUnMmcJbrxQojyiZ0hF2DAfVRucli6Rk5G4aS6DGSxft3NqG+UlPfMxmf6rFa9ubD/7xurIdXbsKDKNYYMBMx3jAWRIdk3p0MuESmxdQAZZKbLQkbU0mZNhfKmfrOctlVaFXKjl127iqFWjU9RBZO4BRK4MAl1OAWGtAEBg/wDG/wbkXWk/Vivf5EM1b65xj+yPr4BtWdjcg=</latexit>

G0 = G(23)<latexit sha1_base64="KQWGpZdwUvA0M/EUEAphwNcidxw=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZNNGNCYkLXGIilwQmpFPOQEPnYtsxIRNeQ1dG3fkwvoBvY0EWCv6rr+f/m5z/eLHgStv2l5VZWV1b38hu5ra2d3b38vsHTRUlkmGDRSKSbY8qFDzEhuZaYDuWSANPYMsb3Uz91iNKxaPwXo9jdAM6CLnPGdVm5BZrp9e1XlqqnJ9Nir18wS7bM5FlcOZQgLnqvfxntx+xJMBQM0GV6jh2rN2USs2ZwEmumyiMKRvRAXYMhjRA5aazpSfkxI8k0UMks/fvbEoDpcaBZzIB1UO16E2H/3mdRPtXbsrDONEYMhMxnp8IoiMy7U76XCLTYmyAMsnNloQNqaRMmwvlTH1nsewyNCtlxy47d5VC9WJ+iCwcwTGUwIFLqMIt1KEBDB7gGd7g3YqtJ+vFev2JZqz5n0P4I+vjG9Wejcg=</latexit><latexit sha1_base64="KQWGpZdwUvA0M/EUEAphwNcidxw=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZNNGNCYkLXGIilwQmpFPOQEPnYtsxIRNeQ1dG3fkwvoBvY0EWCv6rr+f/m5z/eLHgStv2l5VZWV1b38hu5ra2d3b38vsHTRUlkmGDRSKSbY8qFDzEhuZaYDuWSANPYMsb3Uz91iNKxaPwXo9jdAM6CLnPGdVm5BZrp9e1XlqqnJ9Nir18wS7bM5FlcOZQgLnqvfxntx+xJMBQM0GV6jh2rN2USs2ZwEmumyiMKRvRAXYMhjRA5aazpSfkxI8k0UMks/fvbEoDpcaBZzIB1UO16E2H/3mdRPtXbsrDONEYMhMxnp8IoiMy7U76XCLTYmyAMsnNloQNqaRMmwvlTH1nsewyNCtlxy47d5VC9WJ+iCwcwTGUwIFLqMIt1KEBDB7gGd7g3YqtJ+vFev2JZqz5n0P4I+vjG9Wejcg=</latexit><latexit sha1_base64="KQWGpZdwUvA0M/EUEAphwNcidxw=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZNNGNCYkLXGIilwQmpFPOQEPnYtsxIRNeQ1dG3fkwvoBvY0EWCv6rr+f/m5z/eLHgStv2l5VZWV1b38hu5ra2d3b38vsHTRUlkmGDRSKSbY8qFDzEhuZaYDuWSANPYMsb3Uz91iNKxaPwXo9jdAM6CLnPGdVm5BZrp9e1XlqqnJ9Nir18wS7bM5FlcOZQgLnqvfxntx+xJMBQM0GV6jh2rN2USs2ZwEmumyiMKRvRAXYMhjRA5aazpSfkxI8k0UMks/fvbEoDpcaBZzIB1UO16E2H/3mdRPtXbsrDONEYMhMxnp8IoiMy7U76XCLTYmyAMsnNloQNqaRMmwvlTH1nsewyNCtlxy47d5VC9WJ+iCwcwTGUwIFLqMIt1KEBDB7gGd7g3YqtJ+vFev2JZqz5n0P4I+vjG9Wejcg=</latexit><latexit sha1_base64="KQWGpZdwUvA0M/EUEAphwNcidxw=">AAAB7nicbZDLTgIxFIbP4A3xhrp00whG3JAZNNGNCYkLXGIilwQmpFPOQEPnYtsxIRNeQ1dG3fkwvoBvY0EWCv6rr+f/m5z/eLHgStv2l5VZWV1b38hu5ra2d3b38vsHTRUlkmGDRSKSbY8qFDzEhuZaYDuWSANPYMsb3Uz91iNKxaPwXo9jdAM6CLnPGdVm5BZrp9e1XlqqnJ9Nir18wS7bM5FlcOZQgLnqvfxntx+xJMBQM0GV6jh2rN2USs2ZwEmumyiMKRvRAXYMhjRA5aazpSfkxI8k0UMks/fvbEoDpcaBZzIB1UO16E2H/3mdRPtXbsrDONEYMhMxnp8IoiMy7U76XCLTYmyAMsnNloQNqaRMmwvlTH1nsewyNCtlxy47d5VC9WJ+iCwcwTGUwIFLqMIt1KEBDB7gGd7g3YqtJ+vFev2JZqz5n0P4I+vjG9Wejcg=</latexit>

Figure 1: For G “ P4 with V pP4q “ tv1, v2, v3, v4u and EpP4q “ tv1v2, v2v3, v3v4u, we haveLG “ t12, 23, 34u. The labelings corresponding to the permutations id, p14qp23q, p23q and p14q aredepicted (left to right) showing the copies G “ Gid “ Gp14qp23q and G1 “ Gp23q “ Gp14q, whereEpGσq “ tvσ´1piqvσ´1pjq | ij P LGu in all cases.

It is important to note that the set of labels LGσ “ tσpiqσpjq | vivj P EpGσqu of theedges of Gσ is the same for all σ P Sn, i.e. LGσ “ LG for all σ P Sn. Moreover, thecorresponding copies of the vertices and edges of G in Gσ are given by their labels: the copyof vertex vi of G is the vertex of Gσ having label i, while the copy of an edge vivj P EpGqis the edge of Gσ having label ij.

We denote by G the complement graph of G, that is, V pGq “ V pGq and EpGq “tuv | u, v P V pGq, uv R EpGqu. Given e P EpGq and e1 P EpGq Y teu, we say that the graphG´ e` e1 is obtained from G by performing the edge-replacement eÑ e1. If G´ e` e1 is agraph isomorphic to G, we say that the edge-replacement eÑ e1 is feasible. Let

RG “ trsÑ kl | G´ vrvs ` vkvl – Gu

be the set of all feasible edge-replacements of G given by their labels. Notice that, sincefeasible edge-replacements are defined by the labels of the edges, any rs Ñ kl P RG repre-sents also a feasible edge-replacement of any copy Gρ, ρ P Sn. Hence, clearly RGρ “ RG forany ρ P Sn.

Given a feasible edge-replacement, rsÑ kl P RG, we will use the following notation

SGprsÑ klq “ tσ P Sn | Gσ “ G´ vrvs ` vkvlu.

4

We will use sometimes the notation eÑ e1 P RG when we do not require to specify theindexes of the vertices involved in the edge-replacement.

Now we can state the following lemma that will establish the ground for how we aregoing to work with the very interesting graph family of the amoebas. We use right to leftnotation for the composition of permutations, that is, σ ˝ ρ P Sn is defined as σpρpxqq forevery x P rns. We omit the symbol “˝” when there is no confusion.

Lemma 2.2. Let G be a graph defined on the vertex set V “ tv1, v2, . . . , vnu and let LG “tij | vivj P EpGqu. For any rs Ñ kl P RG, σ P SGprs Ñ klq and ρ P Sn, we have thefollowing:

(i) EpGσq “ tvivj | ij P pLGztrsuq Y tkluu .

(ii) pLGztrsuq Y tklu “

σ´1piqσ´1pjq | ij P LG(

.

(iii) Gσ ρ “ Gρ ´ e` e1, where e “ vρ´1prqvρ´1psq and e1 “ vρ´1pkqvρ´1plq.

Proof. (i) Since σ P SGprsÑ klq, by definition, we have

EpGσq “ pEpGqztvrvsuq Y tvkvlu

“ ptvivj | ij P LGu ztvrvsuq Y tvkvlu

“ tvivj | ij P pLGztrsuq Y tkluu .

(ii) By (i) and definition of Gσ, we have

tvivj | ij P pLGztrsuq Y tkluu “ EpGσq “

vσ´1piqvσ´1pjq | ij P LG(

,

from which, by taking the set of pairs of sub-indexes, we obtain

pLGztrsuq Y tklu “

σ´1piqσ´1pjq | ij P LG(

.

(iii) We need to prove that the copy of G associated to the composition σ ρ can be ob-tained by applying the edge-replacement e Ñ e1 to Gρ, where e “ vρ´1prqvρ´1psq ande1 “ vρ´1pkqvρ´1plq. Observe that in Gρ the edge e “ vρ´1prqvρ´1psq is labeled with rs whilethe edge e1 “ vρ´1pkqvρ´1plq is labeled with kl. Then with (ii), we deduce

EpGρ ´ e` e1q “

vρ´1piqvρ´1pjq

ˇ

ˇ ij P pLGztrsuq Y tklu(

“

vρ´1pσ´1piqqvρ´1pσ´1pjqq | ij P LG(

“ EpGσ ρq.

To the sake of comprehension, we show a concrete example.

Example 2.3. Let G “ P4 with V pP4q “ tv1, v2, v3, v4u and EpP4q “ tv1v2, v2v3, v3v4u.Then,

RG “ t12 Ñ 12, 23 Ñ 23, 34 Ñ 34, 12 Ñ 14, 23 Ñ 13, 23 Ñ 24, 23 Ñ 14, 34 Ñ 14u.

5

Consider 12 Ñ 14 P RG which corresponds to the feasible edge-replacement v1v2 Ñ v1v4 (seeFigure 2). Observe that the graph G ´ v1v2 ` v1v4 corresponds to Gσ for all σ P SGp12 Ñ14q “ tp24q, p1432qu. Set σ “ p24q. Now we formulate the three items of Lemma 2.2 forthis example. Recall that L “ t12, 23, 34u and so pLzt12uq Y t14u “ t23, 34, 14u. Also notethat σ´1 “ p24q.

(i) EpGσq “ tvivj | ij P pLzt12uq Y t14uu “ tv2v3, v3v4, v1v4u.

(ii) By (i) and definition of Gσ, we have

tv2v3, v3v4, v1v4u “ EpGσq “

vσ´1piqvσ´1pjq | ij P L(

“ tv1v4, v4v3, v3v2u,

thus the sets of pairs of sub-indices coincide

pLzt12uq Y t14u “ t23, 34, 14u “ t14, 43, 32u “

σ´1piqσ´1pjq | ij P L(

.

(iii) To illustrate this item we consider ρ “ p23q, and so ρ´1 “ p23q. We need to showthat the copy of G associated to the composition σ ρ “ p24q ˝ p23q “ p234q is obtainedby applying the edge-replacement e Ñ e1 to Gρ, where e “ vρ´1p1qvρ´1p2q “ v1v3 ande1 “ vρ´1p1qvρ´1p4q “ v1v4. Note that, in Gρ, the edge e is labeled with 12 while theedge e1 is labeled with 14 and see Figure 3.

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G � v1v2 + v1v4 = G(24)<latexit sha1_base64="0rpM69YLwEdHCdCwfzxcW0rrJeg=">AAAB/XicbVDLSsNAFJ3UV62vqDvdBFuhIpYkFHQjFFzUZQX7gDaEyfS2HTp5MDMplFD0Z3Ql6s6f8Af8G6cxC209m3vuPWdgzvEiRoU0zS8tt7K6tr6R3yxsbe/s7un7By0RxpxAk4Qs5B0PC2A0gKakkkEn4oB9j0HbG9/M9fYEuKBhcC+nETg+HgZ0QAmW6uTqR6X6xcS1Jq59no7qdd1Nynb1bFZy9aJZMVMYy8TKSBFlaLj6Z68fktiHQBKGhehaZiSdBHNJCYNZoRcLiDAZ4yF0FQ2wD8JJ0gwz43QQckOOwEj3394E+0JMfU95fCxHYlGbH//TurEcXDkJDaJYQkCURWmDmBkyNOZVGH3KgUg2VQQTTtUvDTLCHBOpCiuo+NZi2GXSsiuWWbHu7GKtmhWRR8foBJWRhS5RDd2iBmoigh7RM3pD79qD9qS9aK8/1pyWvTlEf6B9fAPpXZLJ</latexit><latexit sha1_base64="0rpM69YLwEdHCdCwfzxcW0rrJeg=">AAAB/XicbVDLSsNAFJ3UV62vqDvdBFuhIpYkFHQjFFzUZQX7gDaEyfS2HTp5MDMplFD0Z3Ql6s6f8Af8G6cxC209m3vuPWdgzvEiRoU0zS8tt7K6tr6R3yxsbe/s7un7By0RxpxAk4Qs5B0PC2A0gKakkkEn4oB9j0HbG9/M9fYEuKBhcC+nETg+HgZ0QAmW6uTqR6X6xcS1Jq59no7qdd1Nynb1bFZy9aJZMVMYy8TKSBFlaLj6Z68fktiHQBKGhehaZiSdBHNJCYNZoRcLiDAZ4yF0FQ2wD8JJ0gwz43QQckOOwEj3394E+0JMfU95fCxHYlGbH//TurEcXDkJDaJYQkCURWmDmBkyNOZVGH3KgUg2VQQTTtUvDTLCHBOpCiuo+NZi2GXSsiuWWbHu7GKtmhWRR8foBJWRhS5RDd2iBmoigh7RM3pD79qD9qS9aK8/1pyWvTlEf6B9fAPpXZLJ</latexit><latexit sha1_base64="0rpM69YLwEdHCdCwfzxcW0rrJeg=">AAAB/XicbVDLSsNAFJ3UV62vqDvdBFuhIpYkFHQjFFzUZQX7gDaEyfS2HTp5MDMplFD0Z3Ql6s6f8Af8G6cxC209m3vuPWdgzvEiRoU0zS8tt7K6tr6R3yxsbe/s7un7By0RxpxAk4Qs5B0PC2A0gKakkkEn4oB9j0HbG9/M9fYEuKBhcC+nETg+HgZ0QAmW6uTqR6X6xcS1Jq59no7qdd1Nynb1bFZy9aJZMVMYy8TKSBFlaLj6Z68fktiHQBKGhehaZiSdBHNJCYNZoRcLiDAZ4yF0FQ2wD8JJ0gwz43QQckOOwEj3394E+0JMfU95fCxHYlGbH//TurEcXDkJDaJYQkCURWmDmBkyNOZVGH3KgUg2VQQTTtUvDTLCHBOpCiuo+NZi2GXSsiuWWbHu7GKtmhWRR8foBJWRhS5RDd2iBmoigh7RM3pD79qD9qS9aK8/1pyWvTlEf6B9fAPpXZLJ</latexit><latexit sha1_base64="0rpM69YLwEdHCdCwfzxcW0rrJeg=">AAAB/XicbVDLSsNAFJ3UV62vqDvdBFuhIpYkFHQjFFzUZQX7gDaEyfS2HTp5MDMplFD0Z3Ql6s6f8Af8G6cxC209m3vuPWdgzvEiRoU0zS8tt7K6tr6R3yxsbe/s7un7By0RxpxAk4Qs5B0PC2A0gKakkkEn4oB9j0HbG9/M9fYEuKBhcC+nETg+HgZ0QAmW6uTqR6X6xcS1Jq59no7qdd1Nynb1bFZy9aJZMVMYy8TKSBFlaLj6Z68fktiHQBKGhehaZiSdBHNJCYNZoRcLiDAZ4yF0FQ2wD8JJ0gwz43QQckOOwEj3394E+0JMfU95fCxHYlGbH//TurEcXDkJDaJYQkCURWmDmBkyNOZVGH3KgUg2VQQTTtUvDTLCHBOpCiuo+NZi2GXSsiuWWbHu7GKtmhWRR8foBJWRhS5RDd2iBmoigh7RM3pD79qD9qS9aK8/1pyWvTlEf6B9fAPpXZLJ</latexit>

� = (24)<latexit sha1_base64="vFfYqvxcXE0mVTpu82tb3sx3aw8=">AAAB7nicbZDNTgIxFIXv4B/iH+rSTSOY4IbMEBLdmJC4cYmJIAlMSKdcoKGdGduOCZnwGroy6s6H8QV8GwvOQsGz+nrPaXLPDWLBtXHdLye3tr6xuZXfLuzs7u0fFA+P2jpKFMMWi0SkOgHVKHiILcONwE6skMpA4H0wuZ7794+oNI/COzON0Zd0FPIhZ9TYkV/uaT6S9KpSq5+X+8WSW3UXIqvgZVCCTM1+8bM3iFgiMTRMUK27nhsbP6XKcCZwVuglGmPKJnSEXYshlaj9dLH0jJwNI0XMGMni/TubUqn1VAY2I6kZ62VvPvzP6yZmeOmnPIwTgyGzEesNE0FMRObdyYArZEZMLVCmuN2SsDFVlBl7oYKt7y2XXYV2req5Ve+2VmrUs0Pk4QROoQIeXEADbqAJLWDwAM/wBu9O7Dw5L87rTzTnZH+O4Y+cj29uGo4q</latexit><latexit sha1_base64="vFfYqvxcXE0mVTpu82tb3sx3aw8=">AAAB7nicbZDNTgIxFIXv4B/iH+rSTSOY4IbMEBLdmJC4cYmJIAlMSKdcoKGdGduOCZnwGroy6s6H8QV8GwvOQsGz+nrPaXLPDWLBtXHdLye3tr6xuZXfLuzs7u0fFA+P2jpKFMMWi0SkOgHVKHiILcONwE6skMpA4H0wuZ7794+oNI/COzON0Zd0FPIhZ9TYkV/uaT6S9KpSq5+X+8WSW3UXIqvgZVCCTM1+8bM3iFgiMTRMUK27nhsbP6XKcCZwVuglGmPKJnSEXYshlaj9dLH0jJwNI0XMGMni/TubUqn1VAY2I6kZ62VvPvzP6yZmeOmnPIwTgyGzEesNE0FMRObdyYArZEZMLVCmuN2SsDFVlBl7oYKt7y2XXYV2req5Ve+2VmrUs0Pk4QROoQIeXEADbqAJLWDwAM/wBu9O7Dw5L87rTzTnZH+O4Y+cj29uGo4q</latexit><latexit sha1_base64="vFfYqvxcXE0mVTpu82tb3sx3aw8=">AAAB7nicbZDNTgIxFIXv4B/iH+rSTSOY4IbMEBLdmJC4cYmJIAlMSKdcoKGdGduOCZnwGroy6s6H8QV8GwvOQsGz+nrPaXLPDWLBtXHdLye3tr6xuZXfLuzs7u0fFA+P2jpKFMMWi0SkOgHVKHiILcONwE6skMpA4H0wuZ7794+oNI/COzON0Zd0FPIhZ9TYkV/uaT6S9KpSq5+X+8WSW3UXIqvgZVCCTM1+8bM3iFgiMTRMUK27nhsbP6XKcCZwVuglGmPKJnSEXYshlaj9dLH0jJwNI0XMGMni/TubUqn1VAY2I6kZ62VvPvzP6yZmeOmnPIwTgyGzEesNE0FMRObdyYArZEZMLVCmuN2SsDFVlBl7oYKt7y2XXYV2req5Ve+2VmrUs0Pk4QROoQIeXEADbqAJLWDwAM/wBu9O7Dw5L87rTzTnZH+O4Y+cj29uGo4q</latexit><latexit sha1_base64="vFfYqvxcXE0mVTpu82tb3sx3aw8=">AAAB7nicbZDNTgIxFIXv4B/iH+rSTSOY4IbMEBLdmJC4cYmJIAlMSKdcoKGdGduOCZnwGroy6s6H8QV8GwvOQsGz+nrPaXLPDWLBtXHdLye3tr6xuZXfLuzs7u0fFA+P2jpKFMMWi0SkOgHVKHiILcONwE6skMpA4H0wuZ7794+oNI/COzON0Zd0FPIhZ9TYkV/uaT6S9KpSq5+X+8WSW3UXIqvgZVCCTM1+8bM3iFgiMTRMUK27nhsbP6XKcCZwVuglGmPKJnSEXYshlaj9dLH0jJwNI0XMGMni/TubUqn1VAY2I6kZ62VvPvzP6yZmeOmnPIwTgyGzEesNE0FMRObdyYArZEZMLVCmuN2SsDFVlBl7oYKt7y2XXYV2req5Ve+2VmrUs0Pk4QROoQIeXEADbqAJLWDwAM/wBu9O7Dw5L87rTzTnZH+O4Y+cj29uGo4q</latexit>

Figure 2: With G “ P4 defined as in Figure 1, we perform the feasible edge replacement 12 Ñ 14,obtaining the copy Gσ of G where σ “ p24q.

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G(234) = G⇢ � v1v3 + v1v4<latexit sha1_base64="EFAIRjcjwa0Ci1wC9dn+Z33VFb4=">AAACBXicbZBNS8MwHMZTX+d8q3r0UrYJE3G020AvwsCDHie4F9hKSbN0DUubkqSDUXrWL6MnUW9+Ab+A38as9qCbzyW//J8nkOfvRpQIaZpf2srq2vrGZmGruL2zu7evHxx2BYs5wh3EKON9FwpMSYg7kkiK+xHHMHAp7rmT67nfm2IuCAvv5SzCdgDHIfEIglKNHL1UuXGSar3RPE2vFA25z9LzqWNNncZZdjQrjl42a2YmYxmsHMogV9vRP4cjhuIAhxJRKMTAMiNpJ5BLgihOi8NY4AiiCRzjgcIQBljYSdYlNU48xg3pYyO7/84mMBBiFrgqE0Dpi0VvPvzPG8TSu7QTEkaxxCFSEeV5MTUkM+YrMUaEYyTpTAFEnKhfGsiHHCKpFldU9a3FssvQrdcss2bd1cutZr6IAjgGJVAFFrgALXAL2qADEHgEz+ANvGsP2pP2or3+RFe0/M0R+CPt4xsi25ZJ</latexit><latexit sha1_base64="EFAIRjcjwa0Ci1wC9dn+Z33VFb4=">AAACBXicbZBNS8MwHMZTX+d8q3r0UrYJE3G020AvwsCDHie4F9hKSbN0DUubkqSDUXrWL6MnUW9+Ab+A38as9qCbzyW//J8nkOfvRpQIaZpf2srq2vrGZmGruL2zu7evHxx2BYs5wh3EKON9FwpMSYg7kkiK+xHHMHAp7rmT67nfm2IuCAvv5SzCdgDHIfEIglKNHL1UuXGSar3RPE2vFA25z9LzqWNNncZZdjQrjl42a2YmYxmsHMogV9vRP4cjhuIAhxJRKMTAMiNpJ5BLgihOi8NY4AiiCRzjgcIQBljYSdYlNU48xg3pYyO7/84mMBBiFrgqE0Dpi0VvPvzPG8TSu7QTEkaxxCFSEeV5MTUkM+YrMUaEYyTpTAFEnKhfGsiHHCKpFldU9a3FssvQrdcss2bd1cutZr6IAjgGJVAFFrgALXAL2qADEHgEz+ANvGsP2pP2or3+RFe0/M0R+CPt4xsi25ZJ</latexit><latexit sha1_base64="EFAIRjcjwa0Ci1wC9dn+Z33VFb4=">AAACBXicbZBNS8MwHMZTX+d8q3r0UrYJE3G020AvwsCDHie4F9hKSbN0DUubkqSDUXrWL6MnUW9+Ab+A38as9qCbzyW//J8nkOfvRpQIaZpf2srq2vrGZmGruL2zu7evHxx2BYs5wh3EKON9FwpMSYg7kkiK+xHHMHAp7rmT67nfm2IuCAvv5SzCdgDHIfEIglKNHL1UuXGSar3RPE2vFA25z9LzqWNNncZZdjQrjl42a2YmYxmsHMogV9vRP4cjhuIAhxJRKMTAMiNpJ5BLgihOi8NY4AiiCRzjgcIQBljYSdYlNU48xg3pYyO7/84mMBBiFrgqE0Dpi0VvPvzPG8TSu7QTEkaxxCFSEeV5MTUkM+YrMUaEYyTpTAFEnKhfGsiHHCKpFldU9a3FssvQrdcss2bd1cutZr6IAjgGJVAFFrgALXAL2qADEHgEz+ANvGsP2pP2or3+RFe0/M0R+CPt4xsi25ZJ</latexit><latexit sha1_base64="EFAIRjcjwa0Ci1wC9dn+Z33VFb4=">AAACBXicbZBNS8MwHMZTX+d8q3r0UrYJE3G020AvwsCDHie4F9hKSbN0DUubkqSDUXrWL6MnUW9+Ab+A38as9qCbzyW//J8nkOfvRpQIaZpf2srq2vrGZmGruL2zu7evHxx2BYs5wh3EKON9FwpMSYg7kkiK+xHHMHAp7rmT67nfm2IuCAvv5SzCdgDHIfEIglKNHL1UuXGSar3RPE2vFA25z9LzqWNNncZZdjQrjl42a2YmYxmsHMogV9vRP4cjhuIAhxJRKMTAMiNpJ5BLgihOi8NY4AiiCRzjgcIQBljYSdYlNU48xg3pYyO7/84mMBBiFrgqE0Dpi0VvPvzPG8TSu7QTEkaxxCFSEeV5MTUkM+YrMUaEYyTpTAFEnKhfGsiHHCKpFldU9a3FssvQrdcss2bd1cutZr6IAjgGJVAFFrgALXAL2qADEHgEz+ANvGsP2pP2or3+RFe0/M0R+CPt4xsi25ZJ</latexit>

Figure 3: With G “ P4 defined as in Figure 1, we perform the feasible edge-replacement given by12 Ñ 14 P RG in the copy Gρ of G where ρ “ p23q, obtaining the copy Gσ ρ “ Gp234q.

6

Item (iii) of Lemma 2.2 means that performing a feasible edge-replacement eÑ e1 P RGin a copy Gρ of G yields the copy of G given by the permutation σ ρ, where we can chooseany σ P SGpe Ñ e1q. It now makes sense to consider the group SG generated by thepermutations associated to all feasible edge-replacements, that is, by the set

EG “ď

eÑe1PRG

SGpeÑ e1q.

Thus,SG “ xEGy .

Clearly, SG acts on the set tGρ | ρ P Snu by means of pσ,Gρq ÞÑ Gσρ, where σ P SGand ρ P Sn. Observe that this action represents what happens when a series of feasibleedge-replacements, represented by σ, is performed on a copy Gρ of G: the result is thegraph Gσρ. We shall also note that a trivial edge-replacement, i.e. an edge-replacementrsÑ kl where tr, su “ tk, lu, is always feasible, and AutpGq – tσ P Sn |Gσ “ Gu ď SG.

The following observation is straightforward from the definition of feasible edge-replacementand item piiiq of Lemma 2.2.

Remark 2.4. Let G be a graph defined on the vertex set V “ tv1, v2, . . . , vnu. For anyvi P V , rsÑ kl P RG, σ P SGprsÑ klq and ρ P Sn, we have

degGσρpviq “

$

&

%

degGρpviq ´ 1, if i P tr, suztk, lu

degGρpviq ` 1, if i P tk, luztr, su

degGρpviq, else.

In the next lemma, we discuss the connection between the feasible edge-replacementsof a graph G and those of its complement graph G, concluding that the correspondingassociated groups are the same.

Lemma 2.5. Let G be a graph defined on the vertex set V “ tv1, v2, . . . , vnu. Then,

(i) For any σ P SG, Gσ “ Gσ.

(ii) rsÑ kl P RG if and only if klÑ rs P RG.

(iii) SGprsÑ klq “ SGpklÑ rsq.

(iv) SG “ SG.

Proof. (i) The statement follows from,

EpGσq “ EpKnqztvσ´1piqvσ´1pjq | ij P Lu “ tvσ´1piqvσ´1pjq | ij R Lu “ EpGσq.

(ii) Let rsÑ kl P RG and σ P SGprsÑ klq. Then Gσ “ G´ vrvs ` vkvl “ G´ vkvl ` vrvs.Since Gσ – G, then Gσ – G, and thus we deduce that G´ vkvl ` vrvs – G, implying thatklÑ rs P RG. The converse is analogous.(iii) By what we showed in items (i) and (ii), we have Gσ “ Gσ “ G´ vkvl ` vrvs “ Gτ forany τ P SGpklÑ rsq. It follows that σ P SGpklÑ rsq. Hence, SGprsÑ klq Ď SGpklÑ rsq.

7

The other inclusion is analogous.(iv) By items (ii) and (iii), we have

SG “ xσ | σ P SGprsÑ klq for some rsÑ kl P RGy

“ xσ | σ P SGpklÑ rsq for some klÑ rs P RGy “ SG.

We shall note that the graphs that we have considered are not necessarily connected.Actually, we will work with (non-connected) graphs containing isolated vertices. To finishthis section, we establish important facts related to the feasible edge-replacements in suchgraphs. For a group S ď Sn acting on rns and a subset X Ď rns, we denote by StabSpXq,the stabilizer of S on X, that is

StabSpXq “ tσ P S | σpxq P X for all x P Xu.

Moreover, for σ P Sn and X Ď rns, we denote with σ|X the restriction of σ to X.

Lemma 2.6. Let G “ H Y H 1 be the disjoint union of two graphs H and H 1, whereG has order n and H has order m ă n. Let V pHq “ tv1, v2, . . . , vmu and V pH 1q “tvm`1, vm`2, . . . , vnu. For each σ P SH and τ P SH 1, define ρ “ ρpσ, τq P Sn as

ρpiq “

"

σpiq, if i P rmsτpiq, else,

Then RH , RH 1 Ď RG and SH ˆ SH 1 – tρpσ, τq | σ P SH , τ P SH 1u ď StabSGprmsq.

Proof. That RH , RH 1 Ď RG is easy to see. Let σ P SH and τ P SH 1 . By definition, σ “σqσq´1 ¨ ¨ ¨σ1 for certain σ1, ¨ ¨ ¨ , σq P EH , while τ “ τq1τq1´1 ¨ ¨ ¨ τ1 for certain τ1, ¨ ¨ ¨ , τq1 P EH 1 .Without loss of generality, assume that q ě q1. Define τj “ idSH1 for q1 ` 1 ď j ď q. For1 ď j ď q, let

ρjpiq “

"

σjpiq, if i P rmsτjpiq, else.

Since RH , R1H Ď RG and, for 1 ď i ď q,

Gρj “ pH YH1qρj “ Hσj YH

1τj ,

then ρ1, ¨ ¨ ¨ , ρq P EG. Moreover, ρ “ ρqρq´1 ¨ ¨ ¨ ρ1 and ρ P StabSGprmsq. Since, clearlySH ˆ SH 1 – tρpσ, τq | σ P SH , τ P SH 1u, the latter is a subgroup of StabSGprmsq.

Remark 2.7. In view of Lemma 2.6, we will identify the groups tρpσ, τq | σ P SH , τ P RH 1uand SH ˆ SH 1 and we will use the notation pσ, τq instead of ρpσ, τq. Since we have thatSH ˆ SH 1 ď StabSGprmsq ď SG, we have in particular that SH – SH ˆ xidSH1

y ď SG andthat SH 1 – xidSH

y ˆ SH 1 ď SG. Hence, again in an abuse of notation, we will say that SHand SH 1 are subgroups of SG.

8

3 Local amoebas and global amoebas

In the previous section we define, for any (not necessarily connected) graph G of order n,a subgroup SG of Sn generated by the set of permutations in Sn associated to the feasibleedge-replacements of G. By means of this group, we are ready to define both types ofamoebas.

3.1 Definitions and examples

Definition 3.1. A graph G of order n is called a local amoeba if SG “ Sn. That is,any labeled copy of G embedded in Kn can be reached, from G, by a chain of feasible edge-replacements. On the other hand, a graph G is called global amoeba if there is an integerT ě 0 such that GY tK1 is a local amoeba for all t ě T .

Note that for the concept of global amoeba, which is the one considered already in theliterature [10], we can maintain the image of a graph G embedded in a complete graph KN ,with N “ n` t much larger than |V pGq| “ n traveling via feasible edge replacements fromany given copy of it to any other one.

It is not difficult to convince oneself that, for every n ě 2, a path Pn is both a localamoeba and a global amoeba, while a cycle Cn is neither a local amoeba nor a globalamoeba, for any n ě 4. After developing some theory, we will provide formal argumentsto prove the above facts and, also, to prove rigorously all statements in the next example,in which we exhibit interesting graphs and families of graphs concerning all possibilities ofbeing, or not, a local or a global amoeba. For graph theoretical concepts and terminology,we refer to [33].

Example 3.2.

1. The following graphs are neither local nor global amoebas:

(a) The star K1,k´1 on k vertices, for k ě 4.

(b) Every (non-complete) r-regular graph, for r ě 2.

2. The following graphs are both, local amoeba and global amoebas:

(a) The path Pk on k vertices, for k ě 2.

(b) The graph Cpk, 1q obtained from a cycle on k vertices by attaching a pendantvertex, for k ě 3.

(c) The graph Hn of order n with V pHnq “ A Y B such that, taking q “ tn2 u,A “ tv1, v2, . . . , vqu and B “ tvq`1, vq`2, . . . , vq`rn

2su, where B is a clique, A is

an independent set and adjacencies between A and B are given by vivq`j P EpHnq

if and only if j ď i, where 1 ď i ď q and 1 ď j ď rn2 s, (see Figure 4).

(d) The tree T5 of order 10 depicted in Figure 4.

3. The following graphs are global but not local amoebas:

9

(a) The graph obtained by taking the disjoint union of t copies of a path of order k,tPk, for t ě 2 and k ě 2.

(b) The disjoint union of a path and a cycle of the same order k, PkYCk, for k ě 3.

(c) The graph Gn of odd order n “ 2q`1, for q ě 4, obtained from Hn´1 by attachinga pendant vertex v2q`1 to vertex v2q (see Figure 4).

4. The following graphs are local but not global amoebas:

(a) The graph Kn ´ tK2 obtained from the complete graph Kn by deleting t disjointedges, for t “ 1 and n ě 4, or t ě 2 and n ě 2t` 1.

(b) The graph C`5 obtained from a cycle on five vertices by adding one edge betweentwo diametrical vertices.

(c) The graph Gn Y tK1, with n “ 2q ` 1, q ě 4, and t ě 1.

We point out that some of the statements in Example 3.2 are easy to prove but someothers are not. For instance, consider the graphs depicted in Figure 4: at a first glance, itmay be not clear why Hn and T5 are both global and local amoebas, and why Gn, for nodd, n ě 9, is a global amoeba but not a local amoeba. In the following section, we willgive structural results that will help us understand the families of local and global amoebas,as well as the relationship between them.

T5 v1

v3

v2

v5

v4

v8

v10

v7

v9

v6

T5

vq+1 vq+2 v2q

vq

vq+3Kq

Gnv1 v2 vqv3

Kq

vnHnv1 v2 v3

n = 2q n = 2q + 1

vq+1 vq+2 vq+3 v2q

vq+1 vq+2 v2q

v1 v2 vq

vq+3

v3

vnKq+1

Hnn = 2q + 1

Gnn = 2q + 1

v1 v2 vq−1v3

Kq

vn

vq+1 vq+2 vq+3 v2q−1 v2q

vq

Figure 4: The graphs Hn (odd and even case), T5, and Gn (n odd).

3.2 Structural results

We begin by noticing simple properties.

Proposition 3.3. Let G be a graph of order n.

10

(i) G is a local amoeba if and only if G is a local amoeba.

(ii) If all feasible edge-replacements of G are trivial, then G is a local amoeba if and onlyif G “ Kn or G “ Kn.

Proof. Item (i) follows from item (iv) of Lemma 2.5 and the definition of local amoeba.To prove item (ii), let G be a graph with only trivial feasible edge-replacements, thenEG “ tσ P Sn |Gσ “ Gu – AutpGq. Since, by definition, SG “ xEGy, then SG “ Sn if andonly if AutpGq – Sn, which holds precisely when G “ Kn or G “ Kn.

Next we prove very useful facts concerning the degree sequences of local and globalamoebas.

Proposition 3.4. Let G be a graph of order n with minimum degree δ and maximumdegree ∆. If G is a local amoeba then, for every integer r with δ ď r ď ∆, there is a vertexv P V pGq with degGpvq “ r.

Proof. Let G be a local amoeba with V “ tv1, v2, . . . , vnu and let k, l P rns be such thatdegGpvkq “ δ and degGpvlq “ ∆. Since G is a local amoeba, SG “ Sn, implying that τ “pk lq P SG. Let σ1, ¨ ¨ ¨ , σq P EG be such that τ “ σqσq´1 ¨ ¨ ¨σ1. Now set τi “ σiσi´1 ¨ ¨ ¨σ1,for 1 ď i ď q, and τ0 “ id. In particular, we have τq “ τ . Consider now the sequence

´

degGτipvkq

¯

0ďiďq.

We know the first and the last values of this sequence, namely degGτ0 pvkq “ degGpvkq “ δ

and degGτq pvkq “ degGτ pvkq “ degGpklqpvkq “ degGpvlq “ ∆. If r P tδ,∆u, we are done.

Now, suppose there is an integer r with δ ă r ă ∆ such that degGτipvkq ‰ r for all 0 ď i ď q.

Let j be the first index where degGτjpvkq ě r` 1. Then we have degGτj´1

pvkq ď r´ 1. But,

since Gτj is obtained by performing a feasible edge-replacement in Gτj´1 , by Observation 2.4,we have | degGτj

pvkq´degGτj´1pvkq| ď 1, which is not possible in this case. Hence, we obtain

a contradiction and it follows that there is a j such that degGτjpvkq “ r. Since Gτj – G, it

follows that G has a vertex of degree r.

Proposition 3.5. Let d1 ě d2 ě . . . ě dn be the degree sequence of a global amoeba G oforder n, and let D “ tdi | i P rnsu. Then,

(i) D “ t0u Y r∆s or D “ r∆s where ∆ is the maximum degree of G, and

(ii) for every i P rns, we have di ď n` 1´ i.

Proof. Item (i) follows since, by definition, being G a global amoeba, there is a T ě 0 suchthat G Y tK1 is a local amoeba for every t ě T , particularly if we choose t “ T ` 1 ě 1.Hence, Proposition 3.4 implies that the degree sequence of the graph G Y tK1 cover alldegrees from the smallest, which is 0, to the largest, which is ∆. Therefore, G necessarilysatisfies dn P t0, 1u and D “ t0u Y r∆s or D “ r∆s, as desired.

For (ii), a simple backwards induction works. Indeed, for i “ n, we have dn ď 1 “pn`1q´n. Now, assuming that di ď n`1´ i for some i ě 2, it follows that di´1 ď di`1 ďn` 1´ i` 1 “ n` 1´ pi´ 1q.

11

The next observation will be a useful tool to determine if a graph G is a local amoeba.To continue we need some terminology. Given a subgroup S ď Sn and k P rns, we denoteby Sk the orbit of k by means of the canonical action of S on rns, i.e.

Sk “ tσpkq | σ P Su.

Also, we use StabSpkq “ tσ P S | σpkq “ ku. By the well known fact that the symmetricgroup Sn is generated by xS Y tpi kquy, where S is a transitive subgroup of StabSnpkq forsome k P rns, we have the following observation.

Remark 3.6. Let G be a graph of order n. Then G is a local amoeba if and only if there is ak P rns such that StabSGpkq acts transitively on rnsztku and pj kq P SG for some j P rnsztku.

By means of Remark 3.6, we get the following.

Lemma 3.7. If G is a local amoeba with δpGq P t0, 1u, then GYK1 is a local amoeba.

Proof. LetG be a local amoeba defined on the vertex set V “ tv1, v2, . . . , vnu. IfG “ Kn, weare done by Proposition 3.3 (ii). Hence, in view of Proposition 3.4 and because δpGq P t0, 1u,we can assume that G has a vertex of degree 1, say degGpvnq “ 1. Consider now the graphGYK1 defined on the vertex set V Ytvn`1u. Consider, as in Lemma 2.6, the permutationspσ, idq P SGYK1 , where σ P SG. Moreover,

Sn – SG – xpσ, idq | σ P SGy ď StabSGYK1pn` 1q

which acts transitively on rns. Also note that pn n` 1q P SGYK1 (by means of the feasibleedge-replacement rn Ñ rpn ` 1q where vr is the only neighbor of vn). With the useof Observation 3.6, we conclude that SGYK1 “ Sn`1, implying that G Y K1 is a localamoeba.

The following lemma is quite technical but necessary for the proof of Theorem 3.9, whichis a cornerstone of this paper. It deals with the situation of edge replacements where noisolated vertex is involved or, on the contrary, when isolated vertices are involved. To havea better control of the situation, we will consider a graph G without isolates and add toit t isolated vertices. Graph theoretically these cases are easy to handle, but the proof ofTheorem 3.9 will require us to work with the group action, and so this language is necessary.

Lemma 3.8. Let G be a graph without isolated vertices, and let G˚ “ G Y tK1 for somet ě 1. Let V pGq “ tv1, v2, . . . , vnu, V pG

˚q “ tv1, v2, . . . , vn`tu. We have the followingproperties:

(i) If e Ñ e1 P RG and σ P SG˚pe Ñ e1q, then σ P StabSG˚ prnsq and σ|rns P SGpe Ñ e1q.In particular, AG˚ ď StabSG˚ prnsq and AG “ tϕ|rns | ϕ P AG˚u.

(ii) If σ P EG˚ is such that with σprq “ k for some r P rns and k P rn` 1, n` ts, then

‚ either σ “ ϕ ˝ pr kq for some ϕ P EG˚ X StabSG˚ prnsq,

‚ or σ “ ϕ ˝ pr kqps lq for some s P rnsztru, l P rn` 1, n` tsztku and ϕ P AG˚.

12

Proof. (i) Let e “ rs and e1 “ kl. As e Ñ e1 P RG, we have r, s, k, l P rns. Since G has noisolates, it follows that σ P StabSG˚ prnsq. Then, by definition,

G˚σ “ G˚ ´ vrvs ` vkvl “ pG´ vrvs ` vkvlq YX “ Gσ|rns YX,

where X “ tvn`i | i P rtsu. Hence, σ|rns P SGpeÑ e1q. In particular, when e “ e1, it followsthat AG˚ ď StabSG˚ prnsq.

(ii) Since k P rn ` 1, n ` ts, we have degG˚pvkq “ 0. Having that σ P EG˚ , σprq “ k, andG has no isolates, we conclude that degG˚pvrq “ 1 in view of Remark 2.4. Moreover, theremust be some s P rns and some l P rn` tsztku such that σ P SG˚prsÑ klq.Suppose first that l P rns. Observe that pr kq P SG˚prsÑ ksq Ď EG˚ , and thus EpG˚

pr kqq “

EpG˚ ´ vrvs ` vkvsq. Therefore,

EpG˚σq “ EpG˚ ´ vrvs ` vlvkq

“ EppG˚ ´ vrvs ` vsvkq ´ vsvk ` vlvkq

“ E´

G˚pr kq ´ vsvk ` vlvk

¯

.

But in the copy G˚pr kq the vertex vk has label r, thus G˚σ “ G˚ϕ˝pr kq, where ϕ P SG˚psr Ñ

lrq Ď EG˚ . Finally, since r, s, l P rns, item (i) yields that ϕ P StabSG˚ prnsq.Now suppose that l P rn ` 1, n ` ts. In this case we have degG˚pvkq “ degG˚pvlq “ 0, andby Remark 2.4 it follows that degG˚pvrq “ degG˚pvsq “ 1. Hence,

EpG˚σq “ EpG˚ ´ vrvs ` vlvkq “ E´

G˚pr kqps lq

¯

.

It follows now that there is a ϕ P AG˚ such that σ “ ϕ ˝ pr kqps lq.

Now we are ready to prove a theorem that gives equivalent statements for the definitionof a global amoeba.

Theorem 3.9. Let G be a non-empty graph defined on the vertex set V “ tv1, v2, . . . , vnu.The following statements are equivalent:

(i) G is a global amoeba.

(ii) For each x P rns, there is a y P SGx such that degGpvyq “ 1.

(iii) GYK1 is a local amoeba.

Proof. We will show (i) ñ (ii) ñ (iii) ñ (i).

To see (i) ñ (ii), let G be a global amoeba. We will first handle the case that G has noisolates and then the case with isolates will easily follow.Case 1: Suppose that G has no isolates.By definition of global amoeba, there is a t ě 1 such that G˚ “ GY tK1 is a local amoeba,that is, SG˚ – Sn`t. Let V pG˚q “ V Y tvn`1, ..., vn`tu and let x P rns. Take a permutationτ P SG˚ with τpxq “ m for some m P rn` 1, n` ts. We know that τ “ σqσq´1 ¨ ¨ ¨σ1 where

13

σ1, ¨ ¨ ¨ , σq P EG˚ . Set τi “ σi . . . σ1, 1 ď i ď q. If q “ 1, we have that τ P EG˚ , and thefacts that x P rns and τpxq P rn`1, n` ts imply that, after an edge-replacement, the vertexvx gets degree 0, meaning that it originally had degree 1. As clearly x P SGx, in this casethere we are done. In case q ě 2, we will show that τ can been chosen having the followingproperties:

(a) τipxq P rns for all 1 ď i ď q ´ 1.

(b) τipxq ‰ τjpxq for any pair i, j with 1 ď i ă j ď q.

(c) σi P StabSG˚ prnsq, for 1 ď i ď q ´ 1.

If τipxq P rn ` 1, n ` ts for some i ă q, then we can take τi instead of τ . Hence, wemay assume property (a). If τipxq “ τjpxq for some pair 1 ď i ă j ď q, then we cantake τ 1 “ σqσq´1 . . . σj`1σiσi´1 . . . σ1 instead of τ . Hence, we may assume (b). Supposeσj R StabSG˚ prnsq for some j P t1, 2, . . . , q ´ 1u. Choose j such that it is minimum withthis property. Then σjprq P rn ` 1, n ` ts for some r P rns, say σjprq “ k. By (a),we have r, k ‰ τj´1pxq. By Lemma 3.8, either there is a ϕ P EG˚ X StabSG˚ prnsq suchthat σj “ ϕ ˝ pr kq, or there is a ϕ P AG˚ such that σj “ ϕ ˝ pr kqps lq for certains P rnsztru, l P rn ` 1, n ` tsztku. Suppose we have the first case. Since ϕ P EG˚ , andτjpxq “ σjpτj´1pxqq “ pϕ ˝ pr kqqpτj´1pxqq “ ϕpτj´1pxqq, we can replace τ by τ 1 “ σ1q . . . σ

11,

with σ1i “ σi for 1 ď i ď q, i ‰ j, and σ1j “ ϕ. Suppose now that σj “ ϕ ˝ pr kqps lq forcertain s P rnsztru, l P rn ` 1, n ` tsztku, where ϕ P AG˚ . Observe that, by item (i) ofLemma 3.8, ϕ P EG˚ X StabSG˚ prnsq. Then σjpsq “ ϕ ˝ pr kqps lq “ ϕplq P rn ` 1, n ` ts.Thus, by (a), we know that s ‰ τj´1pxq. Hence, we can proceed completely analogousto the previous case replacing σ1j by ϕ. Thus, we can assume that σi P StabSG˚ prnsq, for1 ď i ď q ´ 1 and property (c) is satisfied.Set y “ τq´1pxq. Now, since σq P EG˚ , y P rns, and

σqpyq “ σqpτq´1pxqq “ τqpxq “ τpxq “ m P rn` 1, n` ts,

we have, in view of Remark 2.4, that degG˚pvyq “ 1. Finally, we will show that y P SGx. Tothis aim, we will use the fact that σi P StabSG˚ prnsq for each 1 ď i ď q´1 and so σi|rns P SGby Lemma 3.8 (i). Then τq´1|rns

“ pσq´1 . . . σ1q|rns P SG, and we have

τq´1|rnspxq “ τq´1pxq “ y,

implying that y P SGx. Since degGpvyq “ degG˚pvyq “ 1, we have finished.Case 2: Suppose that G has a non-empty set X of isolated vertices.Observe that, by definition and in view of Lemma 3.7, G ´ X is global amoeba if andonly if G is a global amoeba. Hence, if G is a global amoeba, it follows with Case 1 thatpG´Xq YK1 is a local amoeba, and, applying Lemma 3.7 as many times as necessary, weobtain that G is a local amoeba and thus SG “ Sn. As G is non-empty, it has to have atleast one vertex of degree 1 by Lemma 3.5, say vy. Hence, we have that y P SGx “ Snx “ rnsfor every x P rns.

To see (ii) ñ (iii), let G˚ “ GYK1 with V pG˚q “ V Ytvn`1u. Assume, without loss ofgenerality, that v1, . . . , vp are the vertices of degree 1 in G. Note that item (ii) is equivalent

14

to say that rns “ Ypi“1SGi. For every σ P SG, consider, as in Lemma 2.6, the permutationspσ, idq P StabSG˚ pn ` 1q. Let k P SGi for some i P rps. Then there is a σ P SG such thatσpkq “ i. Moreover pi n`1q P SG˚ for every i P rps because of the feasible edge-replacementsiiÑ sipn` 1q P RG˚ , where vsi is the unique neighbor if vi in G. Then

pσ, idq´1pi n` 1qpσ, idqpjq “

$

&

%

n` 1, if j “ kk, if j “ n` 1j, else.

Hence, pσ, idq´1pi n` 1qpσ, idq “ pk n` 1q P SG˚ . Since this holds for each k PŤpi“1 SGi “

rns, pk n` 1q P SG˚ for all k P rns and we conclude that SG˚ – Sn`1. Hence, G˚ “ GYK1

is a local amoeba.

Finally, the implication (iii) ñ (i) is direct by the definition of global amoeba andLemma 3.7.

Theorem 3.9 contains important information that we would like to point out moreclearly. On the one hand, it is shown in item (iii) that the definition of global amoeba canbe reduced to check if a graph together with an isolated vertex is a local amoeba. Thisconfirms the following fact.

Corollary 3.10. Let G be a graph of order n. Then G is a global amoeba if and only if,for any N ě n ` 1 and any two copies F and F 1 of G in KN , F can be transformed intoF 1 by a chain of feasible edge-replacements.

On the other hand, in view of Remark 2.4, it can be deduced from the proof of implication(i) ñ (ii) that we actually have the following fact, which will be a practical tool whenshowing that a graph is a global amoeba or not.

Corollary 3.11. A graph G is a global amoeba if and only if for each x P rns such thatdegGpvxq ě 2, there is a σ P SG such that degGpvσpxqq “ degGpvxq ´ 1.

To see the usefulness of this corollary, see for instance the proof of Example 3.2 3(c)given in Section 6.

Observe also that repeated applications of Theorem 3.9 and Lemma 3.7, together withProposition 3.3 (i) yield the following corollary.

Corollary 3.12. A graph G is a global amoeba if and only if GY tK1 is a local amoeba forany t ě 1 if and only if GY tK1 is a local amoeba for any t ě 1.

To conclude this section, we analyze the relationship between local and global amoebas.Recall that, by Proposition 3.5, every global amoeba G has δpGq P t0, 1u. On the otherhand, a local amoeba can have minimum degree arbitrarily large (see Example 3.2 item 4(a)). Interestingly, a local amoeba with minimum degree 0 or 1 is a global amoeba too,and the converse is true only when δpGq “ 0. We will prove the latter facts in the nextcorollary. Before this, we shall note that, in item 3 of Example 3.2, connected as well asnon-connected global amoebas with minimum degree one are presented which, in fact, arenot local amoebas.

15

Corollary 3.13. Let G be a graph with minimum degree δ.

(i) If δ “ 1 and G is a local amoeba, then G is a global amoeba.

(ii) If δ “ 0, then G is a local amoeba if and only if G is a global amoeba.

Proof. If G is a local amoeba with δ P t0, 1u, then GYK1 is a local amoeba by Lemma 3.7.Hence, it follows with Theorem 3.9 that G is a global amoeba. This yields item (i) and theif-part of item (ii). For the only-if part of (ii), let G be a global amoeba with an isolatedvertex v. By definition, G ´ v is a global amoeba and thus, by Theorem 3.9, G is a localamoeba.

4 Constructions

In this section, we give some constructions of global amoebas that arise from smaller ones.In particular, we will be able to construct large global amoebas, as well as global amoebashaving any connected graph as one of its components and global amoeba-trees with arbi-trarily large maximum degree. In view of Corollary 3.12, every construction given here thatyields a global amoeba G can also be used to construct a local amoeba when consideringthe graph GY tK1 for any t ě 1, which is, in fact, connected.

4.1 Unions and expansions

Arising as a consequence of Theorem 3.9, we will show in the first place that the vertexdisjoint union of two global amoebas is again a global amoeba.

Proposition 4.1. Let H and H 1 be two vertex-disjoint global amoebas. Then G “ H YH 1

is a global amoeba, too.

Proof. Let V pHq “ tv1, v2, . . . , vmu and V pH 1q “ tvm`1, vm`2, . . . , vnu. Let IH and IH 1 bethe sets of all indexes of the vertices of degree one in H and H 1, respectively. Since H andH 1 are global amoebas, we have, by the equivalence of items (i) and (ii) of Theorem 3.9,that

ď

iPIH

SHi “ rms andď

iPIH1

SH 1i “ rm` 1, ns.

Hence, with S ď SG the soubgroup isomorphic to SH ˆ SH 1 (see Lemma 2.6) and I “IH Y IH 1 , we obtain

rns “ď

iPI

Si Ďď

iPI

SGi,

from which, again by the equivalence of items (i) and (ii) of Theorem 3.9, we obtain thatG is a global amoeba.

Observe that the converse statement of Proposition 4.1 is not valid. For example, letH “ Pk and H 1 “ Ck, for k ě 3. The graph G “ H YH 1 is a global amoeba (see item 3 (b)of Example 3.2). However, H “ Ck is not a global amoeba (see item 1 (a) of Example 3.2).

We remark also at this point that there is no corresponding result to Proposition 4.1for local amoebas, since the union of two local amoebas is not necessarily again a local

16

amoeba (see for instance Example 3.2 3.(a)). We present in the following two propositions,the first one giving a union of two vertex-disjoint graphs that is always both a local anda global amoeba, the second one giving a union of several vertex-disjoint global amoebaswhose union is again a global amoeba but never a local amoeba.

Proposition 4.2. Let G be a local amoeba with a vertex v P V pGq such that degpvq “ 1.If H is a copy of G´ v, which is vertex-disjoint from G, then GYH is both, a local and aglobal amoeba.

Proof. Let n “ npGq where V pGq “ tv1, v2, . . . , vnu such that v “ vn, and let V pHq “tvn`1, vn`2, . . . , v2n´1u such that vn`i is the copy of vi, for 1 ď i ď n´ 1. Since G is a localamoeba, we know that SG – Sn. By Lemma 2.6 and Remark 2.7, we have SG ď SGYH .Hence, we can assume that

p1 2 . . . n´ 1q, pn´ 1nq P SGYH .

Let now vj be the neighbor of v and consider the feasible edge-replacement j nÑ n` j n,which gives the permutation

σ “ p1n` 1qp2n` 2q ¨ ¨ ¨ pn´ 1 2n´ 1q P SGYH .

Then we have two permutations p1 2 . . . n´1q and σ which act transitively on r2n´1sztnu.Hence, together with the permutation pn´ 1nq, they generate S2n´1, implying that GYHis a local amoeba. Since it again has a vertex of degree 1, it follows by Corollary 3.10 thatGYH is a global amoeba, too.

Proposition 4.3. Let G1, G2, . . . , Gk be k ě 2 connected and pairwise vertex-disjoint globalamoebas such that epGiq “ epGjq, for 1 ď i, j ď k. Then G “

Ťki“1Gi is a global amoeba

but not a local amoeba.

Proof. Let n “ npGq. By Proposition 4.1, G is a global amoeba. However, G is not a localamoeba because the only possible feasible edge replacements can just interchange edgeswithin one of the components, implying that SG – SG1 ˆ SG2 ˆ . . .ˆ SGt fl Sn.

The next theorem allows us to enlarge a global amoeba by means of taking a copy of aportion of its components where either an edge is added or deleted.

Theorem 4.4. Let G “ H 1 YH2 be a global amoeba, where H 1 and H2 are vertex-disjointsubgraphs of G (where H2 can be possibly empty, meaning that G “ H 1) and such thatEpGq “ EpH 1q Y EpH2q. Let H be a copy of H 1 which is vertex disjoint from G. Then wehave the following facts.

(i) For any e P EpHq, GY pH ` eq is a global amoeba.

(ii) For any e P EpHq, GY pH ´ eq is a global amoeba.

Proof. We will give only the proof of item (i) as the one of (ii) can be deduced similarly.Let V pGq “ tv1, v2, . . . , vnu, and V pH 1q “ tv1, v2, . . . , vmu, where m ď n. Let V pHq “tvn`1, vn`2, . . . , vn`mu and e “ vn`jvn`k P EpHq for some j, k P rms. Assume, withoutloss of generality, that vn`i is the copy of vi in H, 1 ď i ď m. Then pn ` jq pn ` kq Ñ j k

17

is a feasible edge-replacement in G Y pH ` eq and the permutation σ P Sn`m defined byσpiq “ n` i and σpn` iq “ i, for 1 ď i ď m, and σpiq “ i for m` 1 ď i ď n, is contained inSGYpHèqppn` jq pn` kq Ñ j kq. Since SG ď SGYpHèq, we also have SGi Ď SGYpHèqi forany i P rns. If, in particular, i P rms, then n ` i P SGYpHèqi as σpiq “ n ` i. Since G is aglobal amoeba, we know by the equivalence of items (i) and (ii) of Theorem 3.9, that SGi,and thus SGYpHèqi, contains an element l P rns such that degGYpHèqpvlq “ degGpvlq “ 1.Hence, GY pH ` eq is a global amoeba and we are done.

The converse statements of this theorem are not valid. For item (i), we can take pCk YK1qYCpk, 1q that is a global amoeba by Theorem 4.4 (ii) because Cpk, 1q is a global amoeba(Example 3.2 2(b)). However, Ck YK1 is not a global amoeba because it has only trivialfeasible edge-replacements and it is nor complete nor empty (Proposition 3.3 (ii)). On theother hand, for item (ii), consider the graph Ck Y Pk, for k ě 3 that we know is a globalamoeba by Example 3.2 3(c). However, we also know that Pk is a global amoeba, while Ckis not.

Observe that Proposition 4.1 and Theorem 4.4 offer a wide range of possibilities forbuilding amoebas with a diversity of components. For example, given a global amoeba G,the union of G together with any union of graphs that arise from G by adding an edge or bydeleting an edge is a global amoeba. One can also include components that are built fromsmaller components by joining them with edges (needing possibly to apply Theorem 4.4(i)several times). In fact, by iteratively applying Theorem 4.4, one can manage to have anyconnected graph G as a connected component of a global amoeba, as we will show in thefollowing corollary (see also Figure 5 for an illustrative drawing of the method).

Corollary 4.5. Let G be any connected graph. Then there is a global amoeba H having Gas one of its components.

Proof. We will construct a global amoeba H by means of the following recursion. LetH0 “ K1. For i ě 1, we do the following. If Hi´1 fl G, then either there is one edgee P EpHi´1q such that the graph Hi´1 ` e is contained in G as a subgraph, or there isone edge e P EpHi´1 YK1qzEpHi´1q such that pHi´1 Y K1q ` e is contained in G as asubgraph. In the first case we set Hi to be a copy of Hi´1 ` e, in the second case to be acopy of pHi´1 YK1q ` e. Since we add in each step a new edge and the obtained graph isalways contained in G as a subgraph, after m “ |EpGq| steps, we will obtain a componentHm – G. By means of m consecutive applications of Theorem 4.4 (i) (where sometimesHi´1 and sometimes Hi´1YH0 plays the role of H 1) and, since H0 “ K1 is a global amoeba,it follows that H “

Ťmi“0Hi is a global amoeba having one of its components isomorphic to

G.

18

Figure 5: Example illustrating the proof of Corollary 4.5 with G “ K4.

As a consequence of Theorem 4.4, we obtain that there are global amoebas having arbi-trarily large chromatic and clique number, but in proportion to their order these numbersmay be small. In Section 5, we will present an example of a connected global and localamoeba whose clique and chromatic numbers equal to half its order plus one and we showthat this is best possible.

4.2 Fibonacci amoeba-trees

As we know, paths, the simplest trees one can imagine having only 1 and 2-degree ver-tices, are global amoebas. In this section, we will construct an infinite family of treesvia a Fibonacci-recursion which are global amoebas and which will have arbitrarily largemaximum degree (and by Proposition 3.5 vertices of all other possible degrees).

Lemma 4.6. Let G be a graph on vertex set V “ tvi | i P rnsu. Let G “ G1 Y G2 fortwo subgraphs G1 and G2 with respective vertex sets V 1 and V 2. Let J 1 and J2 be the setsof indexes of the vertices in V 1 and V 2, respectively, and let I “ J 1 X J2. If there is aσ P EG1 X

Ş

jPI StabSG1 pjq, then the permutation

pσpiq “

"

σpiq, for i P J 1zIi, for i P J2

is in EG.

Proof. Let σ P EG1 XŞ

jPI StabSG1 pjq. Then there is a feasible-edge replacement rsÑ kl PRG1 with r, s, k, l P J 1. This edge-replacement gives a copy G1σ of G1 that leaves the verticesvi with i P I untouched, i.e. σpiq “ i for all i P I. Then G “ G1 Y G2 – G1σ Y G2 “ G

pσ.Hence, rsÑ kl is also a feasible edge-replacement in G and pσ P SGprsÑ klq Ď EG.

Example 4.7. The graph G depicted below in Figure 6 is built by the union of the graphG1 with index set J 1 “ t1, 2, 3, 4, 5, 6u and the graph G2 with index set J2 “ t4, 5, 6, 7, 8, 9u.The edge-replacement 12 Ñ 13 is feasible in G1 and we have that σ “ p2 3q P SG1p12 Ñ 13q.Since σ “ p2 3q P EG1X

Ş

i“4,5,6 StabSG1 piq, it follows by previous lemma that pσ “ p2 3q P EG.

19

Figure 6: Sketch for Example 4.7.

Let G be a graph on vertex set V “ tvi | i P rnsu and H another graph provided with aspecial vertex called the root of H. Let I “ ti1, i2, . . . , iku Ď rns. We define G ˚I H as thegraph obtained by taking G and k different copies H1, H2, . . . ,Hk of H and identifying theroot of Hj with vertex vij of G, for 1 ď j ď k (see Figure 7).

Lemma 4.8. Let G be a graph on vertex set V “ tvi | i P rnsu and H another graph oforder m provided with a root. Let I “ ti1, i2, . . . , iku Ď rns, N “ n ` kpm ´ 1q and letrN s “ rns Y

Ťk`“1 Ji` be a partition of rN s such that |Ji` | “ m ´ 1 for all 1 ď ` ď k. Let

G˚IH consist of G and copies Hi1 , Hi2 , . . . ,Hik of H such that V pHi`q “ tvi | i P Ji`Yti`uu,for 1 ď ` ď k. For any `, `1 P rks, ` ‰ `1, let ϕi`,i`1 : Ji` Ñ Ji`1 be the bijection given by anisomorphism between Hi` and Hi`1 that sends vi` to vi`1 . If σ P EG X StabSGpIq, then thepermutation

rσpiq “

"

σpiq, for i P rnsϕi`,σpi`qpiq, for i P Ji` , ` P rks

is in EG˚IH .

Proof. Let σ P EG X StabSGpIq. Then there is a feasible-edge replacement rs Ñ kl P RGwith r, s, k, l P rns. This edge-replacement gives a copy Gσ of G such that σpiq P I for alli P I. Then

pG ˚I Hqrσ “ Gσ ˚I H – G ˚I H,

implying that rsÑ kl is also a feasible edge-replacement in G˚IH and thus rσ P EG˚IH .

Example 4.9. Let G “ v1v2v3v4v5 – P5 and H – K1,3 ` e, i.e. a star on three peakstogether with an edge joining two of the vertices of degree 1, where we designate one of thevertices of degree 2 as the root of H. Let I “ t2, 3u, J2 “ t2, 6, 7, 8u, and J3 “ t3, 9, 10, 11u.Then 4 5 Ñ 1 5 P RG with σ “ p1 4qp2 3q P SGp4 5 Ñ 1 5q P EG X StabSGpt2, 3uq, and ϕ2,3 “

ϕ3,2 “ p2 3qp6 9qp7 10qp8 11q. It follows by Lemma 4.8 that rσ “ p1 4qp2 3qp6 9qp7 10qp8 11q PEG˚IH .

20

Figure 7: Sketch of a graph G ˚I H and of Example 4.9.

We will describe a family of trees that are constructed via a Fibonacci recursion. Wedefine T1 “ T2 “ K2. For i ě 2, we define Ti`1 as the tree consisting of one copy T ofTi´1 and one copy T 1 of Ti, where a vertex of maximum degree of T is joined to a vertex ofmaximum degree of T 1 by means of a new edge, see Figure 8. Observe that ∆pTiq “ i ´ 1for i ě 2, while npTiq “ 2Fi, being Fi the i-th Fibonacci number. Note also that, for i ě 4,Ti has only one vertex of maximum degree, which we will call the root of Ti. For the casethat i ď 3, we will designate one of the vertices of maximum degree as the root of Ti andthis will be the vertex that will be used to attach the new edge in the construction of Ti`1.

Figure 8: Fibonacci amoeba-trees Ti, 1 ď i ď 6.

Theorem 4.10. Ti is a global amoeba for all i ě 1.

Proof. Let T be a tree isomorphic to Ti. Let J be the set of indexes of the vertices of T ,i.e. V pT q “ tvk | k P Ju and let c P J such that vc has maximum degree in T . We will showby induction on i that there is a subset S Ď ET X StabST pcq such that xSy acts transitivelyon Jztcu.

If i “ 1, 2, there is nothing to prove. If i “ 3, then T – P4, say T “ v4v3v1v2

with c “ 1. Then the feasible edge-replacements 34 Ñ 24 and 13 Ñ 14 give respectivelythe permutations p2 3q and p3 4q, which act transitively on t2, 3, 4u “ Jztcu. If i “ 4,then let T be the tree built from the path v4v3v1v2 – T3 and a T2 – K2, given by v5v6,

21

and the edge v1v5 joining both trees. Clearly, the only maximum degree vertex is v1 andthus c “ 1. Then the feasible edge-replacements 34 Ñ 24 and 13 Ñ 14 give respectivelythe permutations p2 3q and p3 4q, which together with the automorphism p3 5qp4 6q, acttransitively on r5szt1u “ Jztcu leaving c “ 1 fixed.

Now suppose that i ě 4 and that we have proved the above statement for integervalues at most i. Let T – Ti`1. Hence, |J | “ 2Fi`1. For a subset X Ă J , we defineVX “ tvx | x P Xu and TX “ T rVXs. Let J “ U Y W be a partition of J such thatTU – Ti´1 and TW – Ti. Further, let U “ AY B and W “ C YD be partitions such thatTA – Ti´3, TB – Ti´2, TC – Ti´1, and TD – Ti´2. By construction, vc is the root of TC .Let a, b, d P J be such that va, vb, vd are the roots of TA, TB, and TD, respectively. Noticethat vavbvcvd is a path of length 4 in T . See Figure 9 for a sketch.

Figure 9: Sketch of the tree T – Ti`1 with its subtrees TU – Ti´1 and TW – Ti, and subsubtreesTA – Ti´3, TB – Ti´2, TC – Ti´1, and TD – Ti´2.

By the induction hypothesis, there are subsets SU Ď ETU X StabSTU pbq and SW Ď

ETW X StabSTW pcq such that xSUy acts transitively on Uztbu and xSW y acts transitively on

W ztcu. Let pSU “ tpσ | σ P SUu and pSW “ tpσ | σ P SW u with pσ as in Lemma 4.6. Then, byprecisely this lemma, pSU , pSW Ď ET . Moreover, the transitive action is inherited, i.e., xpSUyacts transitively on Uztbu and xpSW y acts transitively on W ztcu.

Consider now the tree T pB,Dq that is obtained by identifying all vertices from VB withvertex vb and all vertices from VD with vertex vd, i. e. we contract the sets VB and VDeach to a single vertex (see Figure 10). Observe that abÑ ad is a feasible edge-replacementin T pB,Dq with τ “ pb dq P ST pB,Dqpab Ñ adq, and that T – T pB,Dq ˚tb,du Ti´2. SinceTB – TD – Ti´2, there is a bijection ϕ : B Ñ D given by an isomorphism between TBand TD such that ϕpbq “ d. Then, by Lemma 4.8, we have that ab Ñ ad is a feasibleedge-replacement in T with rτ P ST pabÑ adq such that

rτpiq “

$

&

%

ϕpiq, for i P Bϕ´1piq, for i P Di, else,

and which fulfills that rτ P ET . Moreover, rτ leaves c fixed and so rτ P ET X StabST pcq. Nowwe define

S “ pSU Y pSW Y trτu.

22

Since xpSUy acts transitively on Uztbu, and xpSW y acts transitively on W ztcu, these two setstogether with rτ generate a group xSy that acts transitively on Jztcu.

Hence, we have shown that if T – Ti, for any i ě 1, then there is a subset S Ď

ET X StabST pcq such that xSy acts transitively on Jztcu, where c P J such that vc hasmaximum degree in T .

To finish the proof, we will show that there is a permutation rρ P ST such that xSYtrρuyacts transitively on J , meaning that ST acts transitively on J , too, which, by means ofTheorem 3.9(ii), implies that T is a global amoeba. Since we know already that xSy actstransitively on Jztcu, we just need to find a rρ P ST with rρpcq ‰ c. Indeed, there is such apermutation rρ, namely one produced by the feasible edge-replacement cdÑ bd in T , which,by Lemma 4.8, can be obtained by means of the permutation pb cq P ST pU,Cqpcd Ñ bdqthrough

rρpiq “

$

&

%

ψpiq, for i P Cψ´1piq, for i P Ui, else,

where ψ : C Ñ U is the bijection with ψpcq “ b given by an isomorphism between TC andand TU that sends vc to vb. Hence, Ti is a global amoeba for all i ě 1.

Figure 10: Trees T pB,Dq and T pU,Cq.

5 Extremal global amoebas with respect to size, chromaticnumber and clique number

We denote by epGq, χpGq and ωpGq the size (number of edges), the chromatic number(smallest number of colors in a proper vertex coloring) and the clique number (order of amaximum clique) of G respectively.

We shall note that the degrees constraint established in Propositions 3.4 and 3.5 com-promises the number of edges that a global amoeba or a local amoeba with small minimumdegree can have. In this section, we will show that a graph of order n that is a globalamoeba cannot have more than tn

2

4 u edges. Interestingly, it turns out that this bound issharp. We will also prove that the chromatic number, and thus the clique number, of aglobal amoeba of order n can not be greater than tn2 u`1. Again, this upper bound is sharpand we will prove that it is reached when having the maximum possible number of edges.

23

The family of graphs that proves the sharpness in the upper bounds mentioned inthe previous paragraph is Hn, which was given in item 2(c) of Example 3.2 as the graphof order n with V pHnq “ A Y B such that, taking q “ tn2 u, A “ tv1, v2, . . . , vqu andB “ tvq`1, vq`2, . . . , vq`rn

2su, where B is a clique, A is an independent set and adjacencies

between A and B are given by vivq`j P EpHnq if and only if j ď i, where 1 ď i ď qand 1 ď j ď rn2 s. Observe that degpviq “ i for all 1 ď i ď q and degpvq`jq “ n ´ j forall 1 ď j ď rn2 s. Hence, we have one vertex from each degree between 1 and n ´ 1 withexception of vertices utn

2u and vrn

2s that have both degree tn2 u. In [7], it is shown that Hn is

the only graph of order n having tdegpvq | v P V pGqu “ rn´ 1s. This fact will be importantto prove the following proposition.

Proposition 5.1. For every n ě 2, Hn is a global and local amoeba with δpGq “ 1,

epHnq “ tn2

4 u and ωpHnq “ tn2 u` 1.

Proof. We first prove that, for n ě 2, Hn is a global and local amoeba with δpGq “ 1.Observe that Hn can be defined recursively in the following way. By definition, H2 – K2,which is the same as H1 YK1. Now we will show that Hn – Hn´1 YK1 for n ě 3. Indeed,this comes from the fact that the set of all degree values in Hn´1YK1 is rn´2sYt0u yieldingthat the set of all degree values in Hn´1 YK1 is tn´1´d |0 ď d ď n´2u “ rn´1s. Hence,Hn – Hn´1 YK1, for each n ě 2.

To show that Hn is a global and a local amoeba, we proceed again by induction on n.H2 – K2 is clearly both a local and a global amoeba. Now we assume that Hn is a localand a global amoeba for some n ě 2. By Lemma 3.7, it follows that Hn Y K1 is a localamoeba. Then also Hn`1 – Hn YK1 is a local amoeba (Proposition 3.3 (i)), and becauseit has minimum degree 1, it is also a global amoeba (Corollary 3.13).

The facts that epHnq “ tn2

4 u and ωpHnq “ tn2 u ` 1 follow easily from the definition ofHn.

The next theorem gives upper bounds for the edge number epGq, the chromatic numberχpGq, and the clique number ωpGq, of a global amoeba with minimum degree 1. We willuse the Powell-Welsh bound on the chromatic number of a graph G [32] (see [8] for analternative proof):

χpGq ď max1ďiďn

mintdi ` 1, iu (1)

where d1 ě d2 ě . . . ě dn is the degree sequence of G.

Theorem 5.2. If G is a global amoeba of order n with minimum degree δpGq “ 1, then

(i) epGq ď tn2

4 u, and

(ii) ωpGq ď χpGq ď tn2 u` 1,

where all bounds are sharp. Moreover, we have the following relations concerning the equal-ities in the above bounds.

(iii) If epGq “ tn2

4 u then ωpGq “ χpGq “ tn2 u` 1, but the converse is not true.

(iv) We have ωpGq “ tn2 u` 1 if and only if χpGq “ tn2 u` 1.

24

Proof. Let d1 ě d2 ě . . . ě dn “ 1 be the degree sequence of a global amoeba G, and letD “ tdi | i P rnsu. By Proposition 3.5, we know that D “ rd1s where d1 P rn´ 1s and that,for every i P rns,

di ď n` 1´ i. (2)

Now we will prove the four items separately.

(i) By inequality (2), the sum of the tn2 u smallest degrees satisfies

nÿ

i“rn2

s`1

di ďnÿ

i“rn2

s`1

n` 1´ i “

tn2

uÿ

i“1

i “1

2

Yn

2

] ´Yn

2

]

` 1¯

. (3)

Let L be the set of the rn2 s vertices having the largest degrees (corresponding to thedegrees d1, d2, . . . , drn

2s), and let S “ V pGqzL. Denote by epLq the number of edges induced

by the vertices in L and by epL, Sq the number of edges between L and S. Then we have

rn2

sÿ

i“1

di “ 2epLq ` epL, Sq ďQn

2

U ´Qn

2

U

´ 1¯

`

nÿ

i“rn2

s`1

di. (4)

Hence, inequalities (3) and (4) yield

2epGq “nÿ

i“1

di ďQn

2

U ´Qn

2

U

´ 1¯

` 2nÿ

i“rn2

s`1

di

ď

Qn

2

U ´Qn

2

U

´ 1¯

`

Yn

2

] ´Yn

2

]

` 1¯

“

Z

n2

2

^

,

and the bound follows because 12 tn

2

2 u “ tn2

4 u.

(ii) For everyX

n2

\

` 2 ď i ď n, we have, using (2) with i “X

n2

\

` 2, that

mintdi ` 1, iu ď di ` 1 ď dtn2 u`2 ` 1 ďQn

2

U

ď

Yn

2

]

` 1.

For the remaining cases 1 ď i ďX

n2

\

` 1, we obtain as well

mintdi ` 1, iu ď i ďYn

2

]

` 1.

Altogether it follows with (1), that χpGq ď tn2 u ` 1. Finally, the trivial inequality ωpGq ďχpGq yields the result.

(iii) Observe now that a global amoeba G with degree sequence d1 ě d2 ě . . . ě dn “ 1

satisfies epGq “ tn2

4 u if and only if equalities in (3) and (4) hold. This happens if and onlyif, on the one hand, the smallest degrees 1, 2, . . . , tn2 u ´ 1 appear each one once (while thedegree tn2 u appears at least once) and, on the other hand, the sum of the degrees of thern2 s vertices having the largest degrees is exactly

P

n2

T `P

n2

T

´ 1˘

`řni“rn

2s`1 di, meaning that

25

they form a clique and that the complementary set (i.e. the tn2 u vertices of the smallestdegrees) is edge-less. From here, it is easy to see that ωpGq “ tn2 u` 1. Thus, by item piiq,also χpGq “ tn2 u ` 1. To see that the converse is not true, take the graph Gn defined inExample 3.2 3(c) that is a global amoeba (see the proof in Section 6) with minimum degree

1, and ωpGq “ χpGq “ tn2 u` 1, but has epGnq ă tn2

4 u for n ě 4.

(iv) The necessity part is clear because of item (ii). For the converse, suppose that χpGq “tn2 u ` 1. Let L be the set of all vertices of degree at least tn2 u. Since V pGqzL contains allvertices of degree at most tn2 u´ 1, it follows by Proposition 3.5 that

Yn

2

]

´ 1 ď |V pGqzL| “ n´ |L|.

Hence, we obtain that |L| ď rn2 s` 1.We assume first that n is even and we suppose for a contradiction that L is not a clique.

Then we can color the vertices of L with |L| ´ 1 “ n2 different colors such that there are no

adjacent vertices with the same color. Since the vertices in V pGqzL have degree not largerthan n

2 ´ 1, we can proceed coloring the vertices of V pGqzL one after the other by takingalways one of the colors that is not already taken by one of its neighbors. In this way, weuse at most n

2 colors and there are no two adjacent vertices with the same color, implyingthe contradiction χpGq ď n

2 . Hence, L has to be a clique and it follows that ωpGq “ n`12 .

Let now n be odd. Let v P V pGqzL be the vertex of degree n´12 . If |NpvqXL| “ n´1

2 andNpvq X L is a clique, we have finished because then N rvs is a clique and so ωpGq “ n`1

2 “

tn2 u ` 1. Hence, we may assume that Npvq X L is not a clique or that |Npvq X L| ď n´32 .

In both cases we can color the vertices of L with n´12 different colors in such a way that

there is no adjacent pair with the same color but taking also care that there are no morethan n´3

2 colors in Npvq X L. Now we can color v using a color that has been used inLzNpvq. The remaining vertices have degree at most n´3

2 , so that we can proceed in agreedy way as in previous case using no more than n´1

2 colors in total. Hence, it followsthat χpGq ď n´1

2 “ tn2 u, a contradiction.

The search for the extremal family in the bound of item (i) of Theorem 5.2 requiresa much more detailed analysis that takes into account, not only the degree sequence, butthe inner structure of a global amoeba. We already know that, if the graph has maximumdegree n ´ 1, the only extremal graph is Hn (see text before Proposition 5.1). However,if the maximum degree is smaller, there may be different possibilities for the repetitionsamong the higher degrees. Still, we believe that the only possible graph attaining equalityhere is Hn (see Conjecture 7.1).

We finish this section with a simple upper bound on the maximum degree of a globalamoeba with minimum degree 1.

Proposition 5.3. Let G be a global amoeba on n vertices and m edges such that δpGq “ 1.Then

∆pGq ď1

2

`

1`?

1´ 8n` 16m˘

ă 1` 2?m,

and the left inequality is sharp.

26

Proof. Let ∆ “ ∆pGq. By Proposition 3.5, we deduce that 2m ě pn ´∆q `ř∆i“1 i, which

gives 4m ě ∆2 ´∆` 2n. Solving the quadratic inequality, the bound

∆pGq ď1

2

`

1`?

1´ 8n` 16m˘

follows. The bound is sharp for the star forest K1,2YK1,3Y . . .YK1,∆, which can be shownto be a global amoeba by means of Theorem 4.4. Finally, the inequality

1

2

`

1`?

1´ 8n` 16m˘

ă 1` 2?m

is easy to verify.

It is easy to construct graphs, in particular acyclic graphs, satisfying equality in Propo-sition 5.3, namely having n´∆` 1 vertices of degree 1 and the remaining vertices havingdegree 2, 3, . . . ,∆. However, to find constructions of such graphs which are also amoebasis much harder. We leave as an open problem to characterize the family of global amoebasthat attains this bound (see Problem 6 in Section 7).

6 Proofs of the statements in Example 3.2

By applying many of the results obtained in this paper, we will finish proving formally allstatements of Example 3.2. We will use two ways of generating the symmetric group Sn:the one described in Remark 3.6 and the fact that xσ, τy – Sn, where σ is an n-cycle and τany transposition of two consecutive elements of the cycle.

1(a) The star K1,k´1 on k vertices is neither a local amoeba nor a global amoeba for k ě 4since it does not fulfill the conditions of the degrees described in Propositions 3.4 and3.5.

1(b) A non-complete r-regular graph is not a local amoeba by Proposition 3.3 (ii) becausethe regularity implies it has only trivial feasible edge-replacements. An r-regulargraph is not a global amoeba, for r ě 2 in view of Propositions 3.5.

2(a) The path Pk on k vertices is both a local and a global amoeba for k ě 3 becauseof the following. Let Pk be defined on the vertex set tv1, v2, . . . , vku with LpPkq “t1 2, 2 3, . . . , pk ´ 1q ku. Consider the feasible edge-replacements pk ´ 1q k Ñ 1 k and2 3 Ñ 1 3, which give the permutations p1 2 3 . . . kq and p1 2q that generate Sk. Thus,Pk is a local amoeba and, by item (i) of Corollary 3.13, it is also a global amoeba.

2(b) The graph Cpk, 1q obtained from a cycle on k vertices by attaching a pendant vertex,for k ě 3. Let Cpk, 1q be defined on the vertex set tv1, v2, . . . , vk`1u with edgestvivi`1 | i P ZkuYtv1vk`1u. Then 1 pk`1q Ñ k pk`1q and pk´1q k Ñ pk´1q pk`1qare feasible edge-replacements that give the permutations p1 2 3 . . . kq and pk k ` 1qthat generate Sk`1. Thus, Cpk, 1q is a local amoeba and, item (i) of Corollary 3.13,it is also a global amoeba.

27

2(c) The fact that Hn is both a local and a global amoeba is the statement of Proposi-tion 5.1.

2(d) The tree T5 was shown in Theorem 4.10 to be a global amoeba. To see that it is a localamoeba, let T5 have vertices vi, 1 ď i ď 10, distributed as in Figure 11, and considerthe feasible edge-replacements 9 10 Ñ 8 10, 5 6 Ñ 2 6, 1 7 Ñ 1 9, and 1 5 Ñ 7 5 thatproduce the permutations p8 9q, p2 5q, p7 9qp8 10q, and p1 7qp2 8qp3 9qp4 10q. It is notdifficult to check that, these permutations act transitively on r10szt6u (see Figure 11for a visual representation of this partial orbit). Finally, consider the feasible edge-replacement 1 5 Ñ 1 6 that gives the permutation p5 6q, which together with the above4 permutations, generate S10 by Remark 3.6.

3(a) For t ě 2 and k ě 2, tPk is not a local amoeba by Proposition 4.3. However, thegraph tPk is a global amoeba because of Proposition 4.1 and the fact that Pk is a localamoeba (see item 2(a) of this example).

3(b) For k ě 3, the disjoint union of a path Pk and a cycle Ck is a global amoeba by meansof Theorem 4.4(i) because Pk is a global amoeba and Ck can be obtained from Pkby adding an edge. However, it is not a local amoeba. To show this, let us describethe graph with the path v1v2 . . . vk and the cycle vk`1vk`2 . . . v2kvk`1. Since Pk is alocal amoeba, we know that the permutations whose feasible edge-replacements thatinterchange edges with vertices in tv1, v2, . . . , vku generate the symmetric group Sk.Moreover, it is quite simple to check that the permutations involving vertices fromtvk`1, vk`2, . . . , v2ku generate the cyclic group on k elements. Finally, the other pos-sible feasible edge-replacements are those that arise by taking one edge from the cycleand moving it to the path such that we join both end-vertices. These permutationsoperate by interchanging completely the sets t1, 2, . . . , ku and tk ` 1, k ` 2, . . . , 2ku.Thus, we cannot hope for obtaining a copy of Pk Y Ck where both the path and thecycle have vertices from both sets tv1, v2, . . . , vku and tvk`1, vk`2, . . . , v2ku. Hence, itis not a local amoeba.

3(c) Recall that n “ 2q ` 1, where q ě 4. We first note that the degrees of the vertices inGn are the following: degpvjq “ j for 1 ď j ď q, degpvq`jq “ 2q´ j for 1 ď j ď q´ 1,degpv2qq “ q ` 1, and degpvnq “ 1. By Corollary 3.11, to prove that Gn is a globalamoeba, it is enough to show that, for each x P rns such that degpvxq ě 2, there isa σ P SGn such that degpvσpxqq “ degpvxq ´ 1. For 2 ď j ď q, we can see that thefeasible edge replacement j pq` jq Ñ pj´1q pq` jq implies that pj´1 jq P SGn . Also,for 1 ď j ď q ´ 2, the feasible edge replacement j pq ` jq Ñ j pq ` j ` 1q implies thatpq`j q`j`1q P SGn . Finally, the feasible edge replacements pq´1q p2q´1q Ñ pq´1q qand 2q nÑ q n imply, respectively, that pq 2q´1q and pq 2qq belong to SGn . Therefore,every vertex with degree at least 2 can decrease its degree in one unit, as desired. Tosee that Gn is not a local amoeba, observe that SGn contains two orbits, r2qs andtnu, because there is no feasible edge-replacement that can change the role of vn (seeFigure 11 for a for a visual representation).

4(a) The graph G “ Kn ´ tK2 for t “ 1 and n ě 4, or t ě 2 and n ě 2t ` 1, is a localamoeba by Proposition 3.3 (i) since G “ tK2 Y pn´ 2tqK1 is a local amoeba because

28

of Corollary 3.12 and the fact that tK2 is a global amoeba. On the other hand, G isnot a global amoeba by Proposition 3.5 since it is not difficult to see that, in all cases,G has minimum degree at least 2, contradicting Proposition 3.5.

4(b) Let C`5 be defined on the vertex set tv1, v2, v3, v4, v5u with edges v1v2, v2v3, v3v4,v4v5, v1v4. Consider the feasible edge-replacements 4 5 Ñ 2 5 and 1 5 Ñ 3 5 that givepermutations p2 4q and p1 3q respectively. Note that those permutations togetherwith the automorphism p1 4qp2 3q belong to StabS

G`5

p5q. Hence, StabSG`5

p5q acts

transitively on the set r4s and, since we also have p1 5q P SG`5by the feasible edge-

replacement 1 2 Ñ 2 5, we conclude that C`5 is a local amoeba by Remark 3.6. Onthe other hand, C`5 is not a global amoeba because of Proposition 3.5.

4(c) The graph Gn Y tK1, with n “ 2q ` 1, q ě 4, and t ě 1, is a local amoeba becauseof Corollary 3.12 and the fact that Gn is a global amoeba by item 3(c). However,Gn Y tK1 is not a global amoeba because it has minimum degree 2, which contradictsProposition 3.5.

T5 v1

v3

v2

v5

v4

v8

v10

v7

v9

v6

T5

vq+1 vq+2 v2q

vq

vq+3Kq

vq+1 vq+2 v2q

v1 v2 vq

vq+3

v3

v2q+1Kq+1

Hn

Gnv1 v2 vqv3

Kq

vnHnv1 v2 v3

n = 2q + 1

n = 2q n = 2q + 1

vq+1 vq+2 vq+3 v2q

Gnn = 2q + 1

v1 v2 vq−1v3

Kq

vn

vq+1 vq+2 vq+3 v2q−1 v2q

vq

Figure 11: Graph T5 with the transitive action on r10szt6u and the graph Gn with its two orbits.

7 Basic problems about amoebas

In this section, we discuss some problems that arise naturally from the concepts of globaland local amoebas and the theory developed in this paper.

One of our main interests is to find more families of local and/or global amoebas as wellas to develop more methods to construct them. Observe that, besides the Fibonacci-amoebatrees given in Section 4.2, all constructions of global amoebas provided in Section 4 yielddisconnected graphs. Thus, it would be nice to find other constructions that give rise toconnected global amoebas. It would be also interesting to know if there are local or globalamoebas with all possible edge numbers.

Problem 1.

(i) Find other families of global and/or local amoebas. In particular, find other infinitefamilies of connected global amoebas.

29

(ii) Is there a global amoeba on n vertices and m edges for every m with 0 ď m ď tn2

4 u?

(iii) Is there a local amoeba on n vertices and m edges for every m with 0 ď m ď`

n2

˘

?

Of course, the recognition problem and its complexity should be studied. To determineif a graph is a local or a global amoeba, one first has to determine which are its feasibleedge-replacements, a problem that involves checking if two graphs are isomorphic. Theisomorphism problem in graphs has been intensively studied. The best currently acceptedtheoretical algorithm is due to Babai and Luks [4], which has a running time of 2Op

?n lognq

for a graph on n vertices. A quasi-polynomial time algorithm was announced by Babaiin 2015 [2], but its proof is not yet fully peer-reviewed, see [3]. However, there are manygraphs classes in which the isomorphism problem is polynomial [24, 25]. The difference inchecking if a graph G of order n is a local or a global amoeba lies on checking if the groupSG is isomorphic to the symmetric group Sn, or if SG˚ acts transitively on rn ` 1s, whereG˚ “ GYK1 (see Theorem 3.9). Both things can be computed in Op|S|nq-time, given thatS is the set of generators (see [21, 31]).

Problem 2. What is the computational complexity of determining if a graph G is a globaland/or local amoeba?

A structural characterization of the graphs that are global but not local amoebas or ofthose that are local but not global, or of those which are both, that could give clues on howthey can be constructed or recognized may be an interesting problem.

Problem 3. Provide a structural characterization of the following graph families.

(i) Global amoebas that are not local amoebas.

(i) Local amoebas that are not global amoebas.

(i) Graphs that are both, global and local amoebas.

However, as the above problem could be challenging in general, it could be more doableif restricted to a particular class of graphs. In this line, we have studied the Fibonacci-treesTi in Section 4.2 and we have shown that they are global amoebas. We also have shown inExample 3.2 2(d) that T5 is a local amoeba, too, and, while analogous arguments work fori ď 4, it is not clear how to proceed for i ě 6.

Problem 4. Which trees are local/global amoebas? Is the Fibonacci-tree Ti a local amoebafor all i ě 1?

The graph Hn given in Example 3.2 2(c) is shown in Theorem 5.2(i) to have the largestdensity among the global amoebas of minimum degree 1. We believe this is the family thatcharacterizes the equality. We state this as a conjecture.

Conjecture 7.1. If G is a global amoeba of order n and minimum degree 1, then epGq “

tn2

4 u if and only if G – Hn.

For the bound on the chromatic and the clique numbers given in Theorem 5.2(ii), whereHn is also an example for their sharpness, a characterization of the graphs attaining equalitywould be interesting as well.

30

Problem 5. Characterize the families of global amoebas G of order n and minimum degree1 with χpGq “ tn2 u` 1 (and, hence, ωpGq “ tn2 u` 1 by Theorem 5.2(iv)).

The graph Hn is also an example of a global amoeba with the largest possible maximumdegree, namely n´1. We also have shown in Proposition 5.3 that the maximum degree of aglobal amoeba with minimum degree 1 and with m edges is at most 1

2

`

1`?

1´ 8n` 16m˘

,and the bound is attained for the star forest K1,2 YK1,3 Y ¨ ¨ ¨ YK1,∆. However, we do notknow about connected global amoebas attaining the bound. In particular, it is intriguingto discover what is the maximum possible degree of a global amoeba tree. We recall at thispoint that, for the Fibonacci-tree family Ti, i ě 1, that we discussed in Section 4.2, thegrowing rate of the maximum degree of Ti is logarithmical with respect to its order, but itcould be that there are global amoeba trees where the behavior between maximum degreeand order is not that drastic and comes rather closer to

?n.

Problem 6. Let Gn be the family of global amoebas of order n and minimum degree 1.

(i) Characterize the family of all graphs G P Gn such that ∆pGq “ 12

`

1`?

1´ 8n` 16m˘

.

(ii) Determine fpFnq “ maxt∆pF q | F P Fnu for different families Fn Ď Gn, like trees,bipartite graphs, connected graphs, etc.

(iii) In particular for the case of the family Tn of trees on n vertices: is fpTnq “ Θp?nq?

Acknowledgements

We would like to thank BIRS-CMO for hosting the workshop Zero-Sum Ramsey Theory:Graphs, Sequences and More 19w5132, in which the authors of this paper were organizersand participants, and where many fruitful discussions arose that contributed to a betterunderstanding of these topics.

The second author was partially supported by PAPIIT IN111819 and CONACyT project282280. The third author was partially supported by PAPIIT IN116519 and CONACyTproject 282280.

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arxiv:2007.11769v3 [math.co] 25 feb 2021

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