Thin edges in the subgraph induced by noncubic vertices of a brace

Abstract

For a vertex set $X$ in a graph, an edge cut $\partial \left(X\right)$ is the set of edges with exactly one end vertex in $X$. An edge cut $\partial \left(X\right)$ is tight if every perfect matching contains exactly one edge in $\partial \left(X\right)$. A matching covered bipartite graph is a brace if, for every tight cut $\partial \left(X\right)$, $\left|X\right|=1$ or $\left|\overline{X}\right|=1$, where $\overline{X}=V\left(G\right)\setminus X$. As one of the building blocks, braces play an important role in Lovász’s tight cut decomposition of matching covered graphs. An edge $e$ in a brace $G$ is thin if, for every tight cut $\partial \left(X\right)$ of $G-e$, $\left|X\right|\le 3$ or $\left|\overline{X}\right|\le 3$. Carvalho, Lucchesi and Murty conjectured that there exists a positive constant $c$ such that every brace $G$ has $c\left|V\left(G\right)\right|$ thin edges. In this paper, we show that the subgraph induced by nonthin edges of $G$, two end vertices of which are of degree at least 4, is a forest. As an application, we show that every brace with $k\left|V\left(G\right)\right|$ cubic vertices has at least $(2-5k)\left|V\left(G\right)\right|/2+1$ thin edges, where $0<k<0.4$.