Removable edges in cubic matching covered graphs

TL;DR

This paper investigates removable edges in cubic matching covered graphs, showing that such graphs, excluding two specific cases, contain a large matching where every edge is removable, highlighting structural properties of these graphs.

Contribution

It proves that every cubic brick, except for K4 and C6, contains a sizable matching of removable edges, extending understanding of their structural properties.

Findings

Every cubic brick other than K4 and C6 has a matching of size at least |V(G)|/8 with all edges removable.

Identifies structural differences in removable edges among specific classes of matching covered graphs.

Provides bounds on the size of removable-edge matchings in cubic bricks.

Abstract

{ An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is matching covered, which was introduced by Lov\'asz and Plummer in connection with ear decompositions of matching covered graphs. A {\it brick}} is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Improving Lov\'asz's result, Carvalho et al. [Ear decompositions of matching covered graphs, {\em Combinatorica}, 19(2):151-174, 1999] showed that each brick other than $K_4$ and $\overline{C_6}$ has $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we show that every cubic brick $G$ other than $K_4$ and $\overline{C_6}$ has a matching of size at least $|V(G)|/8$, each edge of which is removable in $G$.