THE OVERFULLNESS OF GRAPHS WITH SMALL MINIMUM DEGREE AND LARGE MAXIMUM DEGREE

Given a simple graph G, denote by Delta(G), delta(G), and chi '(G) the maximum degree, the minimum degree, and the chromatic index of G, respectively. We say G is Delta-critical if chi '(G) = Delta(G) + 1 and chi '(H) <= Delta(G) for every proper subgraph H of G, and G is overfull if |E(G)| > Delta(G)(sic)|V (G)|/2(sic). Since a maximum matching in G can have size at most (sic)|V (G)|/2(sic), it follows that chi '(G) = Delta(G) + 1 if G is overfull. Conversely, let G be a Delta-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that G is overfull provided Delta(G) > |V (G)|/3. In this paper, we show that any Delta-critical graph G is overfull if Delta(G) - 7 delta(G)/4 >= (3|V (G)| - 17)/4.