Flows on flow-admissible signed graphs
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 and a 5-flow such that is a nowhere-zero 11-flow, for each edge e, and only if , where is the set of all bridges of the subgraph induced by the edges of .
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 -flow.
A -flow is called balanced if 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 , denote by the set of edges with one end in and the other in . For convenience, we write and for and , 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 be the set of d-vertices in G. The maximum degree of G is denoted by . We use to denote the set of cut-edges of G.
A signed graph 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 If a signed graph is connected and admits a -flow such that contains an even number of negative edges, then it also admits a 3-flow such that [3]
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 -NZF . By Lemma 3.1, G admits a 3-flow such that and if and only if . By Theorem 3.2, G admits a 5-flow such that and Since is a -NZF of G, We are to show that 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 . Then G has two perfect matchings and such that and .
We also need a splitting lemma due to Fleischner [5].
Let G be a graph and v be a vertex. If , we denote by 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 -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 , contracting e is done by deleting e and then (if e is not a loop) identifying its ends. Note that all
References (13)
Nowhere-zero integral flows on a bidirected graph
J. Comb. Theory, Ser. B
(1983)- et al.
Group connectivity of graphs — a nonhomogeneous analogue of nowhere-zero flow properties
J. Comb. Theory, Ser. B
(1992) - et al.
Multiple weak 2-linkage and its applications on integer flows on signed graphs
Eur. J. Comb.
(2018) Nowhere-zero 6-flows
J. Comb. Theory, Ser. B
(1981)- et al.
On flows in bidirected graphs
Discrete Math.
(2005) - et al.
Graph Theory
(2008)
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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).
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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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This research project has been partially supported by National Science Foundation (DMS-1700218).