Thin edges in cubic braces

Abstract

AbstractFor a vertex set in a graph, the edge cut is the set of edges with exactly one end vertex in . An edge cut is tight if every perfect matching of the graph contains exactly one edge in . A matching covered bipartite graph is a brace if, for every tight cut , or , where . Braces play an important role in Lovász's tight cut decomposition of matching covered graphs. The bicontraction of a vertex of degree two in a graph, with precisely two neighbours and , consists of shrinking the set to a single vertex. The retract of a matching covered graph is the graph obtained from by repeatedly the bicontractions of vertices of degree two. An edge of a brace with at least six vertices is thin if the retract of is a brace. McCuaig showed that every brace of order at least six has a thin edge. In a brace of order six or more, Carvalho, Lucchesi and Murty proved that has two thin edges, and conjectured that contains two nonadjacent thin edges. Further, they made a stronger conjecture: There exists a positive constant such that every brace has thin edges. By showing that, in every cubic brace of order at least six, there exists a matching of size at least such that every edge in is thin, we prove that the above two conjectures hold for cubic braces.