Deficiencies of connected regular triangle free graphs

Let G be a finite simple graph having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. In an earlier paper we determined an upper bound for def(G) when G is regular and connected. This upper bound is in general not sharp when G is triangle free. In this paper we study the case when G is triangle free and r-regular. We present an upper bound for defCG) and determine the set of all possible values of def (G) when G is r-regular and (r-2)-edge-connected. 1. INTRODUCTION In this paper the graphs are finite, loopless and have no multiple edges. For the most part our notation and terminology follow Bondy and Murty [2]. Thus G is a graph with vertex set V(G), edge set E(G), v(G) vertices and c(G) edges. However we denote the complement of G by G.