a new family of k-divergent graphs

A new family of K-divergent graphs 1

Martın Matamala a,b,3, Jose Zamora a,2

a Departamento de Ingenierıa Matematica, Facultad de Ciencias Fısicas yMatematicas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile

b Centro de Modelamiento Matematico Universidad de Chile and UMR2071-CNRS, Casilla 170-3, Correo 3, Santiago, Chile

Abstract

A graph G is K-divergent if the sequence |V (Kn(G))| tends to infinity with n,where Kn+1(G) is defined by Kn+1(G) = K(Kn(G)) for n > 0 and K(G) is theintersection graph of the cliques of G. An affine graph is a pair (G, σ) where σ isan automorphism assigning to each vertex of G one of its neighbor. An admissiblecoloring of (G, σ) is a pair (c, π), where c is a vertex coloring of G with colors in aset S and π is a permutation of S satisfying c σ = π c. It is locally bijective ifeach color of S appears exactly once in the closed neighborhood of each vertex ofG. In this work we obtain a structural decomposition of any affine graph (G, σ) interm of the orbits of σ. We also prove the following: if an affine graph (G, σ) hasan admissible locally bijective coloring then G is K−divergent. Using this resultwe construct new families of K-divergent graphs including the cartesian product(C2k+1)k, for any k ≥ 2, where C2k+1 is the cycle of length 2k + 1.

Keywords: K-divergent, expansivity, affine graphs

1 Partially supported by FONDAP on applied Mathematics, Fondecyt 1010442 and Inicia-tiva Cientıfica Milenio ICM P01-005.2 Corresponding author. Email: [email protected] Email: [email protected]

Electronic Notes in Discrete Mathematics 19 (2005) 357–363

1571-0653/2005 Published by Elsevier B.V.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2005.05.048

http://www.elsevier.com/locate/endm

1 Introduction

Let G be a connected graph. The clique graph of G, denoted by K(G), is theintersection graph of the cliques (maximal complete subgraphs) of G. A mainquestion regarding the clique graph operator K, is the study of the behaviorof the iteratively application of K. In this work we provide a new family of K-divergent graphs, that is, graphs such that the iteration K i+1(G) = K(K i(G)),for i ≥ 0, generates a family of graphs whose sizes tend to infinity. Onone hand, a graph can be proved to be K-divergent by computing explicitlyall its iterated clique graphs. This approach was used in [1] to prove theK−divergence of the n-dimensional octahedra On and, later it was used toprove the K−divergence of locally C6 graphs [2] and clockwork graphs [3]. Onthe other hand, it is known that clique divergence is preserved by some kindof morphism (retractions [1], covering [2]). Other notions like expansivity [4]and rank divergence [5] are sufficient conditions for K−divergence.

Our results are related to the notion of expansivity previously mentioned.This notion introduced by Neumann-Lara in [4] was developed in the scopeof coaffine graphs. A coaffination σ of a graph G = ∅ is an automorphismof G such that for every u ∈ V (G), u = σ(u) and σ(u) /∈ N(u). A coaffinegraph is a pair (G, σ) where G is a graph and σ is a coaffine automorphismof G. A morphism f : (G, σ) → (H,φ) is admissible if f σ = φ f . WhenG is a stable subgraph of H under φ (φ(G) ⊆ G) we say that (G, σ) is anadmissible subgraph of (H,φ), where σ is the restriction of φ to G. A coaffinegraph (G, σ) is expansive if there exist a sequence n1, n2, . . . of integers, withni → ∞ and a sequence (H1, φ1), (H2, φ2), . . . of coaffine graphs such that:

• Hi = Hi,1 + . . . + Hi,ri, ri → ∞

• For all i, (Hi, φi) is an admissible subgraph of (Kni(G), σniK ) where σni

K isthe canonical coaffination of Kni(G) induced by σ [4].

As previously mentioned, the goal of this work is to provide new examplesof K−divergent graphs by using the theory of expansive graphs. Actually, wewill focus on a specific result that follows from the next theorem.

Theorem 1.1 (Neumann-Lara [4]) If (G, σ) and (H,φ) are coaffine graphssuch that (G + H, σ + φ) is an admissible subgraph of (K(G), σK), then G isexpansive, where σK(Q) = σ(Q).

A crucial point in our work is that we do not deal with a coaffine graphbut with its complement, an affine graph. Since an independent graph I iscoaffine we get the following Proposition.

M. Matamala, J. Zamora / Electronic Notes in Discrete Mathematics 19 (2005) 357–363358

Proposition 1.2 Let (I, φ) be an independent graph with at least two verticestogether with a coaffination, and let (G, σ) be an affine graph. If (G+I, σ+φ)is an admissible subgraph of (K(G), σK), then G is expansive.

In order to use Proposition 1.2 we will proceed as follows. In Section 2 wegive a complete characterization of coaffine graphs in terms of its coaffination.In the best of our knowledge no such characterization is known. In Section 3we present sufficient conditions on (G, σ) to satisfy the inclusion condition ofProposition 1.2. Finally, in Section 4, by using these results we construct newfamilies of expansive graphs.

2 Affine graphs

In this section we give a characterization of coaffine graphs. Instead of con-sidering the graph itself we consider its complement. An affine graph is a pair(G, σ), where G is a graph and σ is an automorphism such that the image ofeach vertex is one of its neighbors. Clearly, (G, σ) is an affine graph if andonly if (G, σ) is a coaffine graph. For an affine graph (G, σ), the orbit of avertex u ∈ V (G) by σ, is the set O(u) = v ∈ V (G) : ∃i ∈ Z, σi(u) = v. Itis clear that each orbit induces a Hamiltonian subgraph of G and that the setof orbits is a partition of the set of vertices of G. We define the orbit index of(G, σ), denoted by η(G, σ), as the number of orbits induced by σ in G. Let nbe an integer n ≥ 2 and let S be a set of element in Zn = 0, 1, . . . , n − 1.The circulant graph Gn(S) of order n is the graph on the set V = Zn wheretwo vertices i and j in V are adjacent if and only if there is s ∈ S such thati = j + s mod n. In the full version of this work we prove that the class ofaffine graphs (G, σ) with η(G, σ) = 1 is exactly the class of circulant graphs.Since each orbit of an affine graph induces an affine graph, a complete struc-tural description of an affine graph can be obtained by describing how differentorbits are connected.

Let A and B two disjoint sets. We denote by [A,B] the set of all subsetsof A ∪ B with exactly one element of A and one element of B. When A andB are disjoint subsets of the vertex set of a graph G, we denote by [A,B]Gthe set of all edges between A and B.

Let (O1, σ1) and (O2, σ2) be two vertex-disjoint circulant graphs and letF ⊆ [O1, O2]. We define the affine-coupling of (O1, σ1) and (O2, σ2) withgenerator F as the graph G := (O1, σ1) ‖F (O2, σ2) where V (G) = V (O1) ∪V (O2) and E(G) is given by E(G) = E(O1) ∪ E(O2) ∪ E(F ) where

E(F ) = σi1(u1), σ

i2(u2) : u1, u2 ∈ F, u1 ∈ O1, u2 ∈ O2i ∈ N

M. Matamala, J. Zamora / Electronic Notes in Discrete Mathematics 19 (2005) 357–363 359

Lemma 2.1 Let (O1, σ1), (O2, σ2) be two circulant graphs and let F ⊆ [O1, O2].Then the graph G = (O1, σ1) ‖F (O2, σ2) is an affine graph.

Conversely, let (G, σ) be an affine graph with η(G, σ) ≥ 2. We say that twoorbits O1, O2 are adjacent if [O1, O2]G = ∅. We prove that the graph inducedby two orbits of G is the affine-coupling of (O1, σ1) and (O2, σ2) where σi isthe restriction of σ to Oi, i = 1, 2.

Lemma 2.2 Let (G, σ) be an affine graph with η(G, σ) ≥ 2. Let O1, O2 be twoadjacent orbits of G. Then there exists a set F ⊆ [O1, O2]G such that the affinegraph induced by the two orbits is the affine-coupling (O1, σ1) ‖F (O2, σ2).

Since the set of orbits of an affine graph is a partition of the vertex set of thegraph, each edge belongs to an affine-coupling of two orbits. Hence a relevantinformation is given by the connections between the orbits. Let (G, σ) an affinegraph, its orbit-graph is the graph Gσ such that each orbit of G is a vertex ofGσ and two vertices are adjacent if and only if their corresponding orbits areadjacent too. We know that each orbit u ∈ V (Gσ) induces a circulant graphGu in G and from Lemma 2.2 we know that each edge e ∈ E(Gσ) is associatedwith a set of edges Fe. Hence, each affine graph G can be completely describedby a triple (Gσ, Cσ, Dσ), where Cσ = Gu : u ∈ V (Gσ) and Dσ = Fe : e ∈E(Gσ). Conversely, for a connected graph H, a family C = Guu∈V (H) ofcirculant graphs and a family D = Fee∈E(H) of generators, we can constructa graph G := Λ(H,C,D) as follows: each vertex u of H is replaced by thecirculant graph Gu. The remaining edges of G are obtained by adding, foreach edge uv ∈ E(H), the edges of the affine coupling Gu ‖Fuv Gv. UsingLemmas 2.1 and 2.2 we get the following characterization of affine graphs.

Theorem 2.3 Let G a connected graph. G is an affine graph if and only ifthere exist a graph H, a family of circulant graphs C = Guu∈V (H) and afamily of generators D = Fee∈E(H) such that G = Λ(H,C,D).

3 Locally bijective coloring

In this section we obtain sufficient conditions on affine graphs to satisfy hypo-thesis of Proposition 1.2. These conditions will lead us to define a new class ofgraphs containing the complement of the n-dimensional octahedra and somecycles known to be clique divergent. Unfortunately, not every graph in thisclass is an affine graph. However, we prove that all of them satisfy the inclusioncondition of Proposition 1.2.

A (proper) vertex coloring c of a graph G = (V,E) is locally bijective (l.b.)


if each color appears exactly once in the closed neighborhood of each vertex.Clearly each monochromatic set defined by a locally bijective coloring is adominating set and it is easy to see that all have the same size. Hence, if cuses p colors, then |G| = rp, where r is the size of any monochromatic set.

Lemma 3.1 Let G be a graph. If G admits a locally bijective coloring with pcolors then.

• |G| = p if and only if G = Kp, the complete graph on p vertices.

• p = 1 if and only if G has no edges.

• p = 2 if and only if G = Or, the complement of the r-dimensional octahedra,with r = |G|

2.

• p = 3 if and only if it is the vertex-disjoint union of cycles of length 3k,k ∈ N.

As we have said a graph admitting a locally bijective coloring is not neces-sarily an affine graph. Our intention to consider them is the following result.

Theorem 3.2 Let G be a K3−free graph. If G admits a locally bijectivecoloring with p colors, 2 ≤ p ≤ G

3, then G + Ip is an induced subgraph of

K(G).

Let (G, σ) be an affine graph. A pair (c, π) is an admissible coloring of(G, σ) if c : V (G) → S is a vertex coloring of G and π is a permutation of Ssuch that c σ = π c. Notice that the permutation π has no fixed pointssince σ is an affination. An interesting property of admissible locally bijectivecolorings is that they are preserved under admissible isomorphism.

Lemma 3.3 Let f be an admissible isomorphism between (G, σ) and (G′, σ′).If (c, π) is an admissible locally bijective coloring of (G, σ) then (c′, π) is anadmissible locally bijective coloring of (G′, σ′) where c′(u) = c(f−1(u)).

From previous result and Proposition 1.2 we get the following corollary.

Corollary 3.4 Let (G, σ) be an affine graph with G a K3−free graph and|G| ≥ 2. If (G, σ) admits an admissible locally bijective coloring (c, π) with at

most |G|3

colors, then (G, σ) is expansive.

4 Construction of new families of graphs K-divergent

In this section, by using Corollary 3.4 we present some families of affine graphswhose complements are expansives.


Let k ≥ 2 and let G be a graph with vertex set V (G) = (V (C2k+1))k. Two

vertices v = (v1, . . . , vk) and v′ = (v′1, . . . , v

′k) are adjacent if and only if v′ − v

belongs to Ω := e1,−e1, e2 + 2e1,−e2 − 2e1, . . . , ek + ke1,−ek − ke1, whereei = (ei

1, . . . , eik), ei

j = 0 for j = i and eii = 1, and the arithmetic is carried out

in (Z2k+1)k.

Let σ be the affination of G given by σ(v) = v + e1 and let c be the vertexcoloring of G given by c(v) = v1. Then (c, π) is an admissible coloring of(G, σ), where π(u) = u + 1 mod 2k + 1. The neighbors of a vertex v arecolored with colors c(v) + 1, c(v) − 1, c(v) + 2, c(v) − 2, . . . , c(v) + k, c(v) − k.Hence c is a locally bijective coloring. From Corollary 3.4 we conclude that(G, σ) is expansive.

The ideas behind the construction of graph G can be used to prove theexpansivity of two families of graphs.

On one hand, let G′ := (C2k+1)k be the Cartesian product of k copies of

the cycle of length 2k +1. Two vertices v and v′ are adjacent in G′ if and onlyif v′ − v belongs to Ω′ := e1,−e1, . . . , ek,−ek. The smallest example in thisfamily is C5 × C5. In [6] it was proved that its complement is expansive.

It can be shown that the function f given by f(v) = v − e1∑k

i=2 ivi isan admissible isomorphism between (G, σ) and (G′, ϕ), where ϕ(v) = v + e1.Since (c, π) is an admissible locally bijective coloring of (G, σ) we concludefrom Lemma 3.3 that (c f−1, π) is an admissible locally bijective coloring for(G′, σ′). This example can be generalized to the Cartesian product G′ := H1×· · · × Hk, where H1, . . . , Hk are cycles whose lengths, n1, . . . , nk respectively,are multiples of 2k + 1 and n1 = 2k + 1. By instance, C9 × C18 × C27 × C36.

On the other hand, the construction of (G, σ) can be extended by using thedecomposition of affine graphs given in Theorem 2.3. The idea is to replace theorbit-graph of (G, σ), denoted by H, by a more general graph. Let k and p betwo integers such that k ≥ 2 and p ≥ 5 whose precise relation will be definedlater. The orbit-graphs we consider are either k−regular bipartite graphs orK3−free 2k−regular graphs. The circulant associated to each vertex of H isthe circulant Gp(Sp), where Sp = −1, 1, when p is odd, and it is −1, 1, p

2,

when p is even. The affination σ′ in Gp(Sp) is given by σ′(i) = i + 1 mod p.Since each vertex of G can be seen as a pair (u, v) with u a vertex in Gp(Sp)and v a vertex in H, the affination in G is given by σ(u, v) = (σ ′(u), v).

The definition of the generator Fe associated to the edge e of H relays onan orientation and a color assigned to e as follows. When H = (A ∪ B,E)is a k−regular bipartite graph the edges are oriented from A to B and theyare colored using any k−edge-coloring of H with colors in 2, . . . , k + 1.When H is a 2k−regular graph, H is splitted in k edge-disjoint 2-factors


H1, . . . , Hk (vertex disjoint union of cycles) and the edges in each 2-factorare oriented cyclically. For each i = 1, . . . , k, the edges of Hi are coloredwith color i + 1. Having defined an orientation and a coloring of the edgesof H, we define the generator Fe associated with the edge e = v, v′ of Hby using a family of pairwise disjoint set Z = Z1, Z2, . . . , Zk as follows.The relation between p and k is now given according to the structure of theorbit graph H. When H is a k−regular bipartite graph the family Z satisfies⋃k

i=1 Zi = 2, 3, . . . , p − 2 \ p2. When H is a 2k−regular graph the family

Z satisfies⋃k

i=1 Zi = 2, 3, . . . , p2 − 1.

The generator Fe is given by Fe = (0, v)(u, v′) : u ∈ Zi where the edgevv′ is oriented from v to v′ and its color is i. In order to prevent the existenceof K3 in G we impose that for all i = 1, . . . , k if s ∈ Zi then s± 1 mod p /∈ Zi.The graph G constructed in this way is a K3−free affine graph. It is notdifficult to see that (c, π), where c is the vertex coloring c(u, v) := u of G andπ is the permutation π(u) = u + 1 mod p is an admissible coloring of (G, σ).In the full version of this work we prove that it is a locally bijective coloring.From Corollary 3.4 we conclude that (G, σ) is expansive.

References

[1] V. Neumann-Lara, On clique-divergent graphs, in: In Problemes Combinatoireset Theorie des Graphes, Orsay, France, 1978, pp. 313–315.

[2] F. Larrion, V. Neumann-Lara, Locally C6 graphs are clique divergent, DiscreteMath. 215 (1-3) (2000) 159–170.

[3] F. Larrion, V. Neumann-Lara, On clique divergent graphs with linear growth,Discrete Math. 245 (1-3) (2002) 139–153.

[4] V. Neumann-Lara, Clique divergence in graphs, in: L. Lovasz, V. T. Sos (Eds.),Algebraic methods in graph theory, Coll. Math. Soc. Janos Bolyai, vol. 25 Szeged,North-Holland, Amsterdam, 1981, pp. 563–569.

[5] F. Larrion, V. Neumann-Lara, M. A. Pizana, Graph relations, clique divergenceand surface triangulations, submitted to Journal of Graph Theory.

[6] C. P. de Mello, A. Morgana, The clique operator on extended P4-sparse graphs,Mat. Contemp. 25 (2003) 33–47, the Latin-American Workshop on Cliques inGraphs (Rio de Janeiro, 2002).


a new family of k-divergent graphs

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