Vizing's 2-Factor Conjecture Involving Large Maximum Degree

Let G be an n-vertex simple graph, and let (G) and (G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that (G)=(G) or (G)+1. Define G to be -critical if (G)=+1 and (H)<(G) for every proper subgraph H of G. In 1965, Vizing conjectured that if G is an n-vertex -critical graph, then G has a 2-factor. Luo and Zhao showed if G is an n-vertex -critical graph with (G)6n/7, then G has a hamiltonian cycle, and so G has a 2-factor. In this article, we show that if G is an n-vertex -critical graph with (G)n/2, then G has a 2-factor.