Thin edges in cubic braces

Abstract

For a vertex set $X$ in a graph, the 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 of the graph contains exactly one edge in $\partial \left(X\right)$. A matching covered bipartite graph $G$ is a brace if, for every tight cut $\partial \left(X\right)$, $\left|X\right|=1$ or $\left|\overline{X}\right|=1$, where $\overline{X}=V\left(G\right)⧹X$. Braces play an important role in Lovász's tight cut decomposition of matching covered graphs. The bicontraction of a vertex $v$ of degree two in a graph, with precisely two neighbours $v_1$ and $v_2$, consists of shrinking the set $\left\{v_1,v,v_2\right\}$ to a single vertex. The retract of a matching covered graph $G$ is the graph obtained from $G$ by repeatedly the bicontractions of vertices of degree two. An edge $e$ of a brace $G$ with at least six vertices is thin if the retract of $G-e$ is a brace. McCuaig showed that every brace of order at least six has a thin edge. In a brace $G$ of order six or more, Carvalho, Lucchesi and Murty proved that $G$ has two thin edges, and conjectured that $G$ contains two nonadjacent thin edges. Further, they made a stronger conjecture: There exists a positive constant $c$ such that every brace $G$ has $c\left|V\left(G\right)\right|$ thin edges. By showing that, in every cubic brace $G$ of order at least six, there exists a matching $M$ of size at least $\left|V\left(G\right)\right|∕10$ such that every edge in $M$ is thin, we prove that the above two conjectures hold for cubic braces.