On disjoint matchings in cubic graphs: maximum 2- and 3-edge-colorable subgraphs

TL;DR

This paper investigates properties of disjoint matchings in cubic graphs, demonstrating how 2-factors extend to maximum 3-edge-colorable subgraphs, establishing bounds on their sizes, and analyzing ratios of maximum matchings.

Contribution

It introduces new bounds and relationships between 2-factors, 3-edge-colorable subgraphs, and maximum matchings in cubic graphs.

Findings

Any 2-factor extends to a maximum 3-edge-colorable subgraph.

Sum of sizes of maximum 2- and 3-edge-colorable subgraphs is at least twice the number of vertices.

The ratio of maximum matching size to the largest matching in edge-disjoint pairs is at most 9/8.

Abstract

We show that any $2-$factor of a cubic graph can be extended to a maximum $3-$edge-colorable subgraph. We also show that the sum of sizes of maximum $2-$ and $3-$edge-colorable subgraphs of a cubic graph is at least twice of its number of vertices. Finally, for a cubic graph $G$, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let $H$ be the largest matching among such pairs. Let $M$ be a maximum matching of $G$. We show that 9/8 is a tight upper bound for $|M|/|H|$.