Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $\omega \geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than ${5/2} \omega - 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} \omega - 1$, then $G$ has a nowhere-zero 5-flow

Flows and bisections in cubic graphs

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.Comment: 14 pages, 6 figures - revised versio

Flows on the join of two graphs

summary:The join of two graphs $G$ and $H$ is a graph formed from disjoint copies of $G$ and $H$ by connecting each vertex of $G$ to each vertex of $H$. We determine the flow number of the resulting graph. More precisely, we prove that the join of two graphs admits a nowhere-zero $3$-flow except for a few classes of graphs: a single vertex joined with a graph containing an isolated vertex or an odd circuit tree component, a single edge joined with a graph containing only isolated edges, a single edge plus an isolated vertex joined with a graph containing only isolated vertices, and two isolated vertices joined with exactly one isolated vertex plus some number of isolated edges