Nowhere-zero 5-flows and (1, 2)-factors

Martın Matamala a,b and Jose Zamora a,1

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 has a nowhere-zero k-flow if there exists an orientation D of the edgesand an integer flow φ such that for all e ∈ D(G), 0 < |φ(e)| < k. A (1, 2)-factor is asubset of the edges F ⊆ E(G) such that the degree of any vertex in the subgraphinduced by F is 1 or 2. It is known that cubic graphs having a nowhere-zero k-flowwith k = 3, 4 are characterized by properties of the cycles of the graph. We extendthese results by giving a characterization of cubic graphs having a nowhere-zero5-flow based on the existence of a (1, 2)-factor of the graph such that the cycles ofthe graph satisfies an algebraic property.

Keywords: nowhere-zero flow factor

1 Introduction

Let G = (V, E) be a simple directed graph without bridges. A flow in G is anowhere-zero k−flow if its values are in {−(k − 1), . . . , k − 1} \ {0}.

It is easy to see that if G has a nowhere-zero k−flow then every directedgraph obtained from G by reversing the orientations of some of its arcs also

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has a nowhere-zero k−flow. This shows that the existence of a nowhere-zerok−flow of G is a property of the underlying undirected graph.

The concept of nowhere-zero k-flow was introduced by Tutte [6] as a re-finement and a generalization for the face coloring problem in planar graphs.In terms of the four-color Theorem it says that every bridgeless planar graphhas a nowhere-zero 4-flow. This result can not be extended to any bridgelessgraph since the Petersen graph has no a nowhere-zero 4-flow. However, in [7]Tutte formulated his famous 5-flow conjecture which still is open:

Conjecture 1.1 Every bridgeless graph admits a nowhere-zero 5-flow

Work about Conjecture 1.1 have focused in properties of a minimal coun-terexample (see [3],[2],[1]) and into the study of structural properties of graphshaving a nowhere-zero 5-flow (see [5]).

The best approximation for Conjecture 1.1 is a result of Seymour [4] wherehe proved that every bridgeless graph has a nowhere-zero 6-flow.

The motivation for studying this conjecture in cubic graphs has two sources.First, it is known that for the Conjecture 1.1 to be true, it is enough to proveit for cubic graphs. Second, for cubic graphs there exist well known charac-terizations of the existence of nowhere-zero k-flow for k = 3, 4. These char-acterizations provide some intuition about the structural properties of cubicgraph admitting nowhere-zero k−flow.

On one hand, Tutte gave the following characterization.

Theorem 1.2 [6] Let G be a cubic graph. G admits a nowhere-zero 3-flow ifand only if G is bipartite

This result can be seen as a parity condition that must satisfy all cyclesof a cubic graph to admit a nowhere-zero 3−flow. The parity condition beingthat each cycle has even length.

On the other hand, for nowhere-zero 4−flow we only need to check thisparity property in the complement of a perfect matching.

Theorem 1.3 Let G = (V, E) be a cubic graph. G has a nowhere-zero 4-flowif and only if G has a perfect matching M such that all the cycles in G − Mhave even length.

This condition can also be formulated in a more algebraic way: for eachcycle C of G, l(C)2|M∩E(C)| ≡ 0 modulo 2, where l(C) is the length of C andE(C) is the set of edges of C. Notice that in this formulation the conditiondepends on M and must be satisfied for all the cycles.

In Section 2 we state Theorem 2.2 which is our main result. It is somehow

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similar to Theorem 1.3 since it characterizes the existence of nowhere-zero5−flow in terms of an algebraic condition that must satisfy all cycles. In ourcase, the algebraic condition is stated in terms of a subset of edges F suchthat each vertex of G is incident with 1 or 2 edges of F . This kind of sets isknown as (1, 2)-factors. As a by product, we obtain in Section 3 a relationbetween nowhere-zero 5-flow and 4-edge-colorings.

It is known by ([7]) that a graph admits a nowhere-zero k-flow if and onlyif it admits a Γ-flow where Γ is an abelian group of cardinality k.

In the following, we will work with nowhere-zero Zk-flows instead of nowhere-zero k-flows and we will consider only cubic graphs.

2 Nowhere-zero Z5-flow and (1, 2)−factors

In this section we give a new characterization of cubic graphs admitting anowhere-zero Z5-flow.

We note that in the group Z5 the flow equation for cubic graphs have thefollowing property:

Lemma 2.1 In the group Z5, x + y + z = 0 if and only if {x, y, z} ∈{{1, 1, 3}, {4, 4, 2}, {2, 2, 1}, {3, 3, 4}}

In order to state our main result, we need some definitions.

Let us assume that G has a nowhere-zero Z5-flow φ. By Lemma 2.1, it iseasy to see that at each vertex at least one edge has flow 1 or 4 and at leastone edge has flow 2 or 3. Let us color the edges having flow 1 or 4 with colorred and the remaining edges with color blue. This splits the set of edges intotwo (1, 2)−factors corresponding to the sets of edges with color red and theset of edges with color blue, which we call the (1, 2)−factors induced by thenowhere-zero Z5−flow φ.

Now let F be a (1, 2)−factor of the graph G = (V, E). We say that twoadjacent edges (sharing a vertex) are F−related if both belong to F or bothbelong to E \ F . Let v be a vertex and let e, e′ and e′′ be its three incidentedges. The F−parity of the pair (e, e′), pF (e, e′), is 2 when e, e′ are F−related.It is 1 when e and e′′ are F−related and it is 3 when e′ and e′′ are F−related.Notice that the F−parity of (e′, e) is the inverse in Z4 of the F−parity of(e, e′) (see Figure 1 where the continuous edges are in F ).

Let−→C = (e1, e2, . . . , en, e1) be a cycle of G. We define the F−parity of−→

C as the sum of the F−parities of (ei, ei+1), for i = 1, . . . , n − 1, plus the

F−parity of (en, e0). Notice that the F -parity of−→C is the additive inverse in

M. Matamala, J. Zamora / Electronic Notes in Discrete Mathematics 30 (2008) 279–284 281

e v e vv

pF (e, e′) = 2 pF (e, e′) = 3pF (e, e′) = 1

e′′ e′′

ee′ e′ e′

e′′

Fig. 1.

Z4 of the F -parity of the←−C = (en, en−1, . . . e1, en), the cycle with the reverse

order. We say that C is F−null if its F−parity of−→C is zero in Z4.

Now we can state our main result which is a characterization of cubicgraphs admitting a nowhere-zero Z5−flow.

Theorem 2.2 Let G = (V, E) be an undirected cubic graph. G admits anowhere-zero Z5-flow if and only if there exists a (1, 2)-factor F such thatevery cycle C of G is F -null

Proof (Sketch)

To proof the forward direction we define F as a (1, 2)−factor induced bya nowhere-zero Z5−flow φ of G. It is not hard to see that the sum of theF−parities over the vertices in a path P of G is zero modulo 4 if and onlyif the sum of φ(e) over the edges not in P incident with the inner vertices ofP is zero modulo 5. In order to prove that each cycle is F−null we use thefact that for each cycle C the sum of φ(e) over the cut defined by C is zeromodulo 5.

To prove the backward implication we take a spanning tree T ⊆ E of Gwith root r and D(E) an orientation of G. We will define a nowhere-zero Z5-flow such that an arc has value 1 or 4 if it is in F and value 2 or 3 if not. We notethat under the previous assumption and by Lemma 2.1, for any vertex v thereare only two possibles solutions for the flow equation of its incident arcs. Letus call ϕv,i, with i = 1, 2 these solutions. We define a function π : V → {1, 2}such that π(r) = 1 and for all arcs e = (u, v) in T, ϕu,π(u)(e) = ϕv,π(v)(e). If atsome arc f = (x, y) not in T, ϕx,π(x)(f) �= ϕy,π(y)(f), it can be shown that thecycle formed by f in T is not F−null. Since for all the arcs its endverticesdefine the same solution, then the function ϕ : D(E) → Z5 such that for alle = (u, v) ∈ D(E), ϕ(e) = ϕu,π(u)(e) is a nowhere-zero Z5-flow of G. �

Note 1 We note that as we only need to check the property for the cyclesdefined by edges not in T , given the (1, 2)−factor F we can check in polynomialtime whether it is induced by a nowhere-zero Z5−flow.

M. Matamala, J. Zamora / Electronic Notes in Discrete Mathematics 30 (2008) 279–284282

3 Nowhere-zero 5-flow and 4-edge coloring

In this section we show that Theorem 2.2 can be thought as a characterizationof 4−edge-colorings which defines nowhere-zero 5-flow.

On one hand, the following Theorem completely characterizes the existenceof nowhere-zero 4-flow in terms of edge-colorings.

Theorem 3.1 [7] Let G be a cubic graph. G has a nowhere-zero 4-flow if andonly if G is 3-edge coloreable.

On the other hand, the following result shows that if Conjecture 1.1 is truethen, for all cubic bridgeless graphs there exists a nowhere-zero 5-flow whosevalues defines or induces a 4-edge-coloring, it means that for any vertex v thevalues of the flow in the edges incident with v are all different. We say that avertex v is bad for a nowhere-zero Z5-flow (ϕ, D) if there exists two edge in itsincident edges with the same flow. A path of G is directed if all the interiorvertices of the path have one ingoing and one outgoing arc. A directed path is(x,−x)-alternating with x ∈ Z5 if the value of the flow in its edges alternatebetween x and −x.

Proposition 3.2 Let G = (V, E) be a cubic graph having a nowhere-zero5-flow φ. Then there exists a nowhere-zero 5-flow φ′ that induces a 4-edge-coloring of G.

Proof. Let (ϕ′, D′) be a nowhere zero 5-flow that minimize the number ofbad vertices. We assume that there exist a bad vertex u with two incidentsarcs with the same flow x. We note that both start in u or both finish inu because the third arc can not be zero. In both cases, let e1 = (u, v) oneof them and let P = e1e2 . . . el the longest directed alternating (x,−x)-pathstarting with e1. Note that all the interior vertex of P are not bad. Then forall e ∈ P we define D′′(e) = −D′(e) and ϕ′′(e) = −x. Note that in Z5, x �= −xand the third arc can not be neither x or −x. For the rest of the arcs e wedefine D′′(e) = D′(e) and ϕ′′(e) = ϕ′(e). Note that for all v ∈ V the flowequation do not change. Then (ϕ′′, D′′) is a nowhere-zero 5-flow with less badvertices than (ϕ′, D′). This contradict the minimality of (ϕ′, D′), then no suchu can exist, so (ϕ′, D′) induce a 4-edge coloring. �

Unfortunately, there are 4-edge-colorings not induced by a nowhere-zero5-flow. In the Figure 2 we show a 4-edge coloring of the Petersen graph whichis not induced by a nowhere-zero Z5-flow.

Nevertheless, Theorem 2.2 gives a condition for 4−edge colorings so as theyare induced by nowhere-zero 5-flow. In fact, each 4−edge-coloring defines six

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Fig. 2.

possibles (1, 2)−factors. If this 4−edge coloring is induced by a nowhere-zero5-flow then, at least one of its associated (1, 2)−factors must satisfies thecondition stated in Theorem 2.2. Then by Note 1 we can check in polynomialtime if a 4-edge coloring is induced by a nowhere-zero 5-flow.

References

[1] Celmins, U., On cubic graphs that do not have an edge-3-coloring, Ph.DThesis, Department of Combinatorics and Optimization, University of Waterloo,Waterloo, Canada (1884).

[2] Kochol, M., Reduction of the 5-flow conjecture to cyclically 6-edge-connectedsnarks, J. Combin. Theory Ser. B 90 (2004), pp. 139–145, dedicated to AdrianBondy and U. S. R. Murty.

[3] Kochol, M., Restrictions on smallest counterexamples to the 5-flow conjecture,Combinatorica 26 (2006), pp. 83–89.

[4] Seymour, P. D., Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30 (1981),pp. 130–135.

[5] Steffen, E., Tutte’s 5-flow conjecture for graphs of nonorientable genus 5, J.Graph Theory 22 (1996), pp. 309–319.

[6] Tutte, W. T., On the imbedding of linear graphs in surfaces, Proc. London Math.Soc. (2) 51 (1949), pp. 474–483.

[7] Tutte, W. T., A contribution to the theory of chromatic polynomials, CanadianJ. Math. 6 (1954), pp. 80–91.

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