On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

TL;DR

This paper proves that determining if a cubic bridgeless graph can be covered by four perfect matchings is NP-complete, constructs an infinite family of such graphs with unique properties, and explores implications for cycle covers.

Contribution

It establishes NP-completeness for the four perfect matchings cover problem and introduces a new infinite family of snarks with specific cycle cover properties.

Findings

Deciding the four perfect matchings cover is NP-complete.

Constructed an infinite family of snarks with no four perfect matchings cover.

Identified a snark with no cycle cover shorter than 4/3 of its edges plus 2.

Abstract

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family $\cal F$ of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known. It turns out that the family $\cal F$ also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with $m$ edges has length at least $\tfrac43m$, and we show that this inequality is strict for graphs of $\cal F$. We also construct the first known snark with no cycle cover of length less than $\tfrac43m+2$.