# On Perfect Matchings in Matching Covered Graphs

## TL;DR

This paper investigates the structure of perfect matchings in matching-covered graphs, providing counterexamples to a previous conjecture and characterizing certain bipartite graphs with equivalent classes.

## Contribution

It constructs infinite families of regular graphs with large equivalent classes that defy previous switching-equivalence conjectures and characterizes bipartite graphs with equivalent classes.

## Key findings

- Existence of infinitely many regular graphs with large equivalent classes not switching-equivalent to trivial sets.
- Counterexample to the conjecture that non-feasible sets are always switching-equivalent to empty or full edge sets.
- Characterization of bipartite graphs with equivalent classes and those with removable edges.

## Abstract

Let $G$ be a matching-covered graph, i.e., every edge is contained in a perfect matching. An edge subset $X$ of $G$ is feasible if there exists two perfect matchings $M_1$ and $M_2$ such that $|M_1\cap X|\not\equiv |M_2\cap X| \pmod 2$. Lukot'ka and Rollov\'a proved that an edge subset $X$ of a regular bipartite graph is not feasible if and only if $X$ is switching-equivalent to $\emptyset$, and they further ask whether a non-feasible set of a regular graph of class 1 is always switching-equivalent to either $\emptyset$ or $E(G)$? Two edges of $G$ are equivalent to each other if a perfect matching $M$ of $G$ either contains both of them or contains none of them. An equivalent class of $G$ is an edge subset $K$ with at least two edges such that the edges of $K$ are mutually equivalent. An equivalent class is not a feasible set. Lov\'asz proved that an equivalent class of a brick has size 2. In this paper, we show that, for every integer $k\ge 3$, there exist infinitely many $k$-regular graphs of class 1 with an arbitrarily large equivalent class $K$ such that $K$ is not switching-equivalent to either $\emptyset$ or $E(G)$, which provides a negative answer to the problem proposed by Lukot'ka and Rollov\'a. Further, we characterize bipartite graphs with equivalent class, and characterize matching-covered bipartite graphs of which every edge is removable.