Covering a cubic graph by 5 perfect matchings

TL;DR

This paper proves the Berge Conjecture for specific classes of cubic graphs, including hypohamiltonian graphs, by demonstrating the existence of five perfect matchings covering all edges.

Contribution

It establishes the Berge Conjecture for cubic graphs with a nearly Hamiltonian circuit and those with a 2-factor of two circuits, expanding known cases.

Findings

Berge Conjecture holds for cubic graphs with a circuit missing one vertex.

Berge Conjecture holds for bridgeless cubic graphs with a 2-factor of two circuits.

The result implies the conjecture for hypohamiltonian cubic graphs.

Abstract

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge Conjecture holds for hypohamiltonian cubic graphs.