Perfect Matching Covers of Cubic Graphs of Oddness 2

Wuyang Sun,Fan Wang

doi:10.37236/7175

Abstract

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

Highlights

Finite and simple graphs are considered in this paper
A perfect matching cover of a graph G is a set of perfect matchings of G such that each edge of G is contained in at least one member of it
Conjecture 1 holds for bridgeless cubic graphs which have no nontrivial 3-edge-cuts and have 3 perfect matchings which miss at most 4 edges

Summary

Introduction

Finite and simple graphs are considered in this paper. A k-factor of a graph G is a spanning k-regular subgraph of G. Conjecture 1 holds for bridgeless cubic graphs which have no nontrivial 3-edge-cuts and have 3 perfect matchings which miss at most 4 edges It is proved by Hou et al [10]. In [19], one author of this paper showed that a cubic graph G with n vertices has a perfect matching cover of order at most 5 if G has a circuit of length n − 1 or has a 2-factor with exactly two circuits. Mazzuoccolo [8] showed that every bridgeless cubic graph G with m edges has 5 perfect matchings which cover at least. Esperet and Mazzuoccolo [2] proved that the problem that deciding whether a bridgeless cubic graph has a perfect matching cover of order at most 4 is NP-complete. If G is a connected bridgeless cubic graph of oddness 2, G has a perfect matching cover of order at most 11

Notations and two technical lemmas

Proof of Theorem 4