Triangle-Free Equimatchable Graphs

TL;DR

This paper characterizes triangle-free equimatchable graphs, extending previous work to include girth at least 4, and provides a linear time recognition algorithm for these graphs.

Contribution

It offers a complete structural characterization of triangle-free equimatchable graphs and extends known results to girth at least 4.

Findings

Characterization of equimatchable graphs with girth at least 4.

Identification of specific triangle-free equimatchable graph families.

Linear time recognition algorithm for these graphs.

Abstract

A graph is called equimatchable if all of its maximal matchings have the same size. Frendrup et al. [8] provided a characterization of equimatchable graphs with girth at least $5$. In this paper, we extend this result by providing a complete structural characterization of equimatchable graphs with girth at least $4$, i.e., equimatchable graphs with no triangle, by identifying the equimatchable triangle-free graph families. Our characterization also extends the result given by Akbari et al. in [1], which proves that the only connected triangle-free equimatchable $r$-regular graphs are $C_5$, $C_7$ and $K_{r,r}$, where $r$ is a positive integer. Given a non-bipartite graph, our characterization implies a linear time recognition algorithm for triangle-free equimatchable graphs.