b -Invariant Edges in Near-Bipartite Bricks

Abstract

A connected graph $G$ is matching covered if it contains at least one edge and every edge lies in some perfect matching of $G$. Lovász proved that every matching covered graph $G$ can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges; we let $b\left(G\right)$ denote the number of bricks. An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is also matching covered. Furthermore, a removable edge $e$ of a brick $G$ is $b$-invariant if $b\left(G-e\right)=1$. Confirming a conjecture of Lovász, de Carvalho, Lucchesi, and Murty proved that every brick, distinct from $K_4$, ${\overline{C}}_6$, and the Petersen graph, has a $b$-invariant edge. A brick $G$ is near-bipartite if it has a pair of edges $R≔\left\{e_1,e_2\right\}$ such that $G-R$ is a bipartite matching covered graph. In this paper, strengthening the result of Zhang et al., we show that in a near-bipartite brick $G$ with $\mid V\left(G\right)\mid \ge 6$, every vertex of $G$, except at most six vertices, is incident with at most two non-$b$-invariant edges. Consequently, $G$ has at least $\frac{\mid V\left(G\right)\mid -6}{2}b$-invariant edges, and all such graphs attaining this lower bound are presented.