Flows on flow-admissible signed graphs

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Abstract

In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero 6-flow. Bouchet himself proved that such signed graphs admit nowhere-zero 216-flows and Zýka further proved that such signed graphs admit nowhere-zero 30-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow.

Introduction

Graphs or signed graphs considered in this paper are finite and may have multiple edges or loops. For terminology and notations not defined here we follow [1], [4], [11].

In 1983, Bouchet [2] proposed a flow conjecture that every flow-admissible signed graph admits a nowhere-zero 6-flow. Bouchet [2] himself proved that such signed graphs admit nowhere-zero 216-flows; Zýka [13] proved that such signed graphs admit nowhere-zero 30-flows. In this paper, we prove the following result.

Theorem 1.1

Every flow-admissible signed graph admits a nowhere-zero 11-flow.

In fact, we prove a stronger and very structural result as follows, and Theorem 1.1 is an immediate corollary.

Theorem 1.2

Every flow-admissible signed graph G admits a 3-flow f1 and a 5-flow f2 such that f=3f1+f2 is a nowhere-zero 11-flow, |f(e)|9 for each edge e, and |f(e)|=10 only if eB(supp(f1))B(supp(f2)), where B(supp(fi)) is the set of all bridges of the subgraph induced by the edges of supp(fi) (i=1,2).

Theorem 1.2 may suggest an approach to further reduce 11-flows to 9-flows.

The main approach to prove the 11-flow theorem is the following result, which, we believe, will be a powerful tool in the study of integer flows of signed graphs, in particular to resolve Bouchet's 6-flow conjecture.

Theorem 1.3

Every flow-admissible signed graph admits a balanced nowhere-zero Z2×Z3-flow.

A Z2×Z3-flow (f1,f2) is called balanced if supp(f1) contains an even number of negative edges.

The rest of the paper is organized as follows: Basic notations and definitions will be introduced in Section 2. Section 3 will discuss the conversion of modulo flows into integer flows. In particular a new result to convert a modulo 3-flow to an integer 5-flow will be introduced and its proof will be presented in Section 5. The proofs of Theorem 1.2, Theorem 1.3 will be presented in Sections 4 and 6, respectively.

Section snippets

Signed graphs, switch operations, and flows

Let G be a graph. For U1,U2V(G), denote by δG(U1,U2) the set of edges with one end in U1 and the other in U2. For convenience, we write δG(U1) and δG(v) for δG(U1,V(G)U1) and δG({v}), respectively. The degree of v is the number of edges incident with v, where each loop is counted twice. A d-vertex is a vertex with degree d. Let Vd(G) be the set of d-vertices in G. The maximum degree of G is denoted by Δ(G). We use B(G) to denote the set of cut-edges of G.

A signed graph (G,σ) is a graph G

Modulo flows on signed graphs

Just like in the study of flows of ordinary graphs and as Theorem 1.3 indicates, the key to make further improvement and to eventually solve Bouchet's 6-flow conjecture is to further study how to convert modulo 2-flows and modulo 3-flows into integer flows. The following lemma converts a modulo 2-flow into an integer 3-flow.

Lemma 3.1

[3]

If a signed graph is connected and admits a Z2-flow f1 such that supp(f1) contains an even number of negative edges, then it also admits a 3-flow f2 such that supp(f1)supp(f

Proof of the 11-flow theorem

Now we are ready to prove Theorem 1.2, assuming Theorem 1.3, Theorem 3.2.

Proof of Theorem 1.2

Let G be a connected flow-admissible signed graph. By Theorem 1.3, G admits a balanced Z2×Z3-NZF (g1,g2). By Lemma 3.1, G admits a 3-flow f1 such that supp(g1)supp(f1) and |f1(e)|=2 if and only if eB(supp(f1)).

By Theorem 3.2, G admits a 5-flow f2 such that supp(f2)=supp(g2) andEf2=±3=.

Since (g1,g2) is a Z2×Z3-NZF of G,supp(f1)supp(f2)=supp(g1)supp(g2)=E(G).

We are to show that f=3f1+f2 is a nowhere-zero 11-flow

Proof of Theorem 3.2

As the preparation of the proof of Theorem 3.2, we first need some necessary lemmas.

The first lemma is a stronger form of the famous Petersen's theorem, and here we omit its proof (see Exercise 16.4.8 in [1]).

Lemma 5.1

Let G be a bridgeless cubic graph and e0E(G). Then G has two perfect matchings M1 and M2 such that e0M1 and e0M2.

We also need a splitting lemma due to Fleischner [5].

Let G be a graph and v be a vertex. If FδG(v), we denote by G[v;F] the graph obtained from G by splitting the edges of F

Proof of Theorem 1.3

In this section, we will complete the proof of Theorem 1.3, which is divided into two steps: first to reduce it from general flow-admissible signed graphs to cubic shrubberies (see Lemma 6.6); and then prove that every cubic shrubbery admits a balanced Z2×Z3-NZF by showing a stronger result (see Lemma 6.13).

We first need some terminology and notations. Let G be a graph. For an edge eE(G), contracting e is done by deleting e and then (if e is not a loop) identifying its ends. Note that all

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1

This research project has been partially supported by National Natural Science Foundation of China (No. 11901318), Natural Science Foundation of Tianjin (No. 19JCQNJC14100) and the Fundamental Research Funds for the Central Universities, Nankai University (No. 63191425).

2

This research project has been partially supported by National Natural Science Foundation of China (No. 11871397), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2020JM-083) and the Fundamental Research Funds for the Central Universities (No. 3102019ghjd003).

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