Vizing's 2-Factor Conjecture Involving Large Maximum Degree

Let G be a simple graph of order n, and let ∆(G) and χ′(G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ′(G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ′(G) = ∆(G) + 1 and χ′(G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic critical, then the independence number α(G) ≤ n/2, where n is the order of G. Furthermore, he conjectured that, in fact, G has a 2-factor. Luo and Zhao showed that if G is an n-vertex edge-chromatic critical graph, then α(G) ≤ n/2 provided that ∆(G) ≥ n/2, and G is hamiltonian (and thus has a 2-factor) if ∆(G) ≥ 6n/7. In this paper, we show that if G is edge-chromatic critical, then G has a 2-factor provided that ∆(G) ≥ n/2.