Intersecting 1-factors and nowhere-zero 5-flows

TL;DR

This paper investigates conditions under which bridgeless cubic graphs admit nowhere-zero 5-flows, focusing on the intersection properties of 1-factors and cyclic edge connectivity.

Contribution

It establishes new sufficient conditions linking 1-factor intersections and cyclic edge connectivity for the existence of nowhere-zero 5-flows in cubic graphs.

Findings

Graphs with cyclic connectivity at least 6 and intersection number at most 2 have nowhere-zero 5-flows.

Graphs with cyclic connectivity at least 5μ₂(G)-3 also have nowhere-zero 5-flows.

Provides new criteria connecting graph structure to flow properties.

Abstract

Let $G$ be a bridgeless cubic graph, and $\mu_2(G)$ the minimum number $k$ such that two 1-factors of $G$ intersect in $k$ edges. A cyclically $n$-edge-connected cubic graph $G$ has a nowhere-zero 5-flow if (1) $n \geq 6$ and $\mu_2(G) \leq 2$ or (2) if $n \geq 5 \mu_2(G)-3$