Characterization Of A Class Of Graphs Related To Pairs Of Disjoint Matchings

TL;DR

This paper characterizes a specific class of graphs where the maximum ratio between the size of a maximum matching and the largest matching in pairs of disjoint matchings is exactly 5/4, revealing structural properties of these graphs.

Contribution

It provides a complete characterization of graphs achieving the ratio of 5/4, including the structure of their spanning subgraphs and minimal components.

Findings

Graphs with ratio exactly 5/4 contain a specific spanning subgraph.

The class of such graphs is characterized by minimal components.

The ratio does not exceed 5/4 for any graph.

Abstract

For a given graph consider a pair of disjoint matchings the union of which contains as many edges as possible. Furthermore, consider the relation of the cardinalities of a maximum matching and the largest matching in those pairs. It is known that this relation does not exceed 5/4 for any graph. We characterize the class of graphs for which this relation is precisely 5/4. Our characterization implies that these graphs contain a spanning subgraph, every component of which is the minimal graph of this class.