Vizing's 2‐Factor Conjecture Involving Large Maximum Degree

Let G be an n ‐vertex simple graph, and let Δ ( G ) andχ ′ ( G )denote the maximum degree and chromatic index of G , respectively. Vizing proved thatχ ′ ( G ) = Δ ( G )or Δ ( G ) + 1 . Define G to be Δ‐critical ifχ ′ ( G ) = Δ + 1 andχ ′ ( H ) < χ ′ ( G )for every proper subgraph H of G . In 1965, Vizing conjectured that if G is an n ‐vertex Δ‐critical graph, then G has a 2‐factor. Luo and Zhao showed if G is an n ‐vertex Δ‐critical graph with Δ ( G ) ≥ 6 n / 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 Δ‐critical graph with Δ ( G ) ≥ n / 2 , then G has a 2‐factor.