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

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

2006-07-04

arxiv.org/abs/math/0607077v1

Mathematics

Combinatorics

Scientific paper

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.

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