Flows and bisections in cubic graphs

TL;DR

This paper investigates k-weak bisections in cubic graphs, establishing that every cubic graph admits a 5-weak bisection and exploring conditions for 4-weak bisections related to perfect matchings.

Contribution

It proves that all cubic graphs have a 5-weak bisection and examines the existence of 4-weak bisections in graphs with perfect matchings.

Findings

Every cubic graph admits a 5-weak bisection.

Cubic graphs with perfect matchings (except Petersen) likely admit 4-weak bisections.

Some cubic graphs without perfect matchings do not admit 4-weak bisections.

Abstract

A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$ vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r \rfloor$-weak bisection. In this paper we study problems related to the existence of $k$-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.