On non-feasible edge sets in matching-covered graphs

Abstract

Let $G=(V,E)$ be a matching-covered graph and $X$ be an edge set of $G$. $X$ is said to be feasible if there exist two perfect matchings $M_1$ and $M_2$ in $G$ such that $|M_1\cap X|\not \equiv|M_2\cap X|\ (\mbox{mod } 2)$. For any $V_0\subseteq V$, $X$ is said to be switching-equivalent to $X\oplus \nabla_G(V_0)$, where $\nabla_G(V_0)$ is the set of edges in $G$ each of which has exactly one end in $V_0$ and $A \oplus B$ is the symmetric difference of two sets $A$ and $B$. Lukot'ka and Rollov\'a showed that when $G$ is regular and bipartite, $X$ is non-feasible if and only if $X$ is switching-equivalent to $\emptyset$. This article extends Lukot'ka and Rollov\'a's result by showing that this conclusion holds as long as $G$ is matching-covered and bipartite. This article also studies matching-covered graphs $G$ whose non-feasible edge sets are switching-equivalent to $\emptyset$ or $E$ and partially characterizes these matching-covered graphs in terms of their ear decompositions. Another aim of this article is to construct infinite many $r$-connected and $r$-regular graphs of class 1 containing non-feasible edge sets not switching-equivalent to either $\emptyset$ or $E$ for an arbitrary integer $r$ with $r\ge 3$, which provides negative answers to problems asked by Lukot'ka and Rollov\'a and He, et al respectively.