1-factor and cycle covers of cubic graphs

TL;DR

This paper investigates the structure of cubic graphs using 1-factors, establishing conditions for cycle covers, Berge-covers, and properties related to girth, while also addressing conjectures and problems in the field.

Contribution

It introduces new structural results on cubic graphs based on 1-factors, providing bounds and conditions for cycle covers, and resolves a problem posed by Zhang.

Findings

A new upper bound for the girth based on $(G)$

Existence of 4-cycle and 5-cycle covers for certain cubic graphs

Counterexample to Zhang's problem

Abstract

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with $\mu_4(G) = 0$ have a 4-cycle cover of length $\frac{4}{3} |E(G)|$ and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang. We also give a negative answer to a problem of Zhang.