On the perfect matching index of bridgeless cubic graphs

TL;DR

This paper investigates the minimal number of perfect matchings needed to cover all edges in bridgeless cubic graphs, exploring the properties of graphs with a perfect matching index of 4 and related conjectures.

Contribution

It introduces the parameter τ(G) for covering edges with perfect matchings and provides insights into graphs with a perfect matching index of 4, relating to longstanding conjectures.

Findings

Defined the parameter τ(G) for edge coverage by perfect matchings.

Analyzed properties of graphs with perfect matching index 4.

Provided information on the class of graphs with index 4.

Abstract

If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings $M_1,...,M_6$ of $G$ with the property that every edge of $G$ is contained in exactly two of them and Berge conjectured that its edge set can be covered by 5 perfect matchings. We define $\tau(G)$ as the least number of perfect matchings allowing to cover the edge set of a bridgeless cubic graph and we study this parameter. The set of graphs with perfect matching index 4 seems interesting and we give some informations on this class.