Ear Decompositions of Matching Covered Graphs

Abstract

different from \(\) and \(\) has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovasz and Plummer establishing the existence of ear decompositions. We also show that every brick G different from \(\) and \(\) has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovasz. We also give a simple proof of another theorem due to Lovasz which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of \(\) or the fourth graph in the sequence is an odd-subdivision of \(\). Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author.