Overfullness of critical class 2 graphs with a small core degree

Abstract

Let G be a simple graph, and let n, ∆(G) and χ ′ (G) be the order, the maximum degree and the chromatic index of G, respectively. We callThe core of G, denoted by G ∆ , is the subgraph of G induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph G with ∆(G) ≥ 4, if the maximum degree of G ∆ is at most two, then G is overfull, which in turn gives ∆(G) > n/2 + 1. We show that for any critical class 2 graph G, if the minimum degree of G ∆ is at most two and ∆(G) > n/2 + 1, then G is overfull.