Characterization of saturated graphs related to pairs of disjoint matchings

TL;DR

This paper investigates the ratio between the maximum matching size in pairs of disjoint matchings and the largest possible matching, providing new characterizations of graphs achieving the ratio of 1 using path and cycle decompositions.

Contribution

It introduces a novel approach using graph decompositions into paths and cycles to analyze and characterize graphs with optimal matching ratios.

Findings

Characterization of graphs with ratio 1 for matchings covered by maximum matchings.

Decomposition into paths and even cycles as a tool for studying matching ratios.

Extension of previous bounds on the ratio of disjoint matchings.

Abstract

We study the ratio, in a finite graph, of the sizes of the largest matching in any pair of disjoint matchings with the maximum total number of edges and the largest possible matching. Previously, it was shown that this ratio is between 4/5 and 1, and the class of graphs achieving 4/5 was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study this ratio. We then use this technique to characterize the graphs achieving ratio 1 among all graphs that can be covered by a certain choice of a maximum matching and maximum disjoint matchings.