Minimal 2-matching-covered graphs

A perfect 2-matching MM of a graph GG is a spanning subgraph of GG such that each component of MM is either an edge or a cycle. A graph GG is said to be 2-matching-covered   if every edge of GG lies in some perfect 2-matching of GG. A 2-matching-covered graph is equivalent to a “regularizable” graph, which was introduced and studied by Berge. A Tutte-type characterization for 2-matching-covered graph was given by Berge. A 2-matching-covered graph is minimal   if G−eG−e is not 2-matching-covered for all edges ee of GG. We use Berge’s theorem to prove that the minimum degree of a minimal 2-matching-covered graph other than K2K2 and K4K4 is 2 and to prove that a minimal 2-matching-covered graph other than K4K4 cannot contain a complete subgraph with at least 4 vertices.