Removable edges in cubic matching covered graphs

Lu Fuliang, Qian Jianguo

Published 2022-02-09Version 1

An edge $e$ in a matching covered graph $G$ is 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 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. Carvalho et al. [Ear decompositions of matching covered graphs, 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 $G$. We show that every cubic brick $G$ other than $K_4$, $\overline{C_6}$ and a particular graph of order 8 has a matching of size at least $|V(G)|/7$, each edge of which is removable in $G$. Moreover, if $G$ is a 3-edge-connected near-bipartite cubic graph, then $G$ has a matching of size at least $|V(G)|/2-3$, each edge of which is removable in $G$. The lower bounds for the two sizes of matchings are sharp.