Nowhere-Zero 3-Flows in Signed Graphs

Yezhou Wu,Cun-Quan Zhang,Dong Ye,Wenan Zang

doi:10.1137/130941687

Abstract

Tutte observed that every nowhere-zero $k$-flow on a plane graph gives rise to a $k$-vertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph $G$ has a face-$k$-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero $k$-flow. However, if the surface is nonorientable, then a face-$k$-coloring corresponds to a nowhere-zero $k$-flow in a signed graph arising from $G$. Graphs embedded in orientable surfaces are therefore a special case that the corresponding signs are all positive. In this paper, we prove that if an 8-edge-connected signed graph admits a nowhere-zero integer flow, then it has a nowhere-zero 3-flow. Our result extends Thomassen's 3-flow theorem on 8-edge-connected graphs to the family of all 8-edge-connected signed graphs. And it also improves Zhu's 3-flow theorem on 11-edge-connected signed graphs.

Highlights

Graphs considered in this paper may have multiple edges and loops unless otherwise stated
Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a k-vertex-coloring of its dual, and vice versa
In response to Zhu’s open question [16], we offer the following conjecture whose validity would imply Tutte’s 3-flow conjecture

Summary

Introduction

Graphs considered in this paper may have multiple edges and loops unless otherwise stated. The following two lemmas are well-known facts (see [10] and [16]) in graph theory, that is, that this flipping operation does not affect the existence of a nowhere-zero integer flow in a signed graph. Let (G, σ) be a signed graph with precisely n(G, σ) negative edges.

Results

Conclusion