Hamiltonian Cycles in Critical Graphs with Large Maximum Degree

It is shown by Luo and Zhao (J Graph Theory 73:469–482, 2013) that an overfull $$\Delta $$Δ-critical graph with n vertices that satisfies $$\Delta \ge \frac{n}{2}$$Δ≥n2 is Hamiltonian. If Hilton’s overfull subgraph conjecture (Chetwynd and Hilton 100:303–317, 1986) was proved to be true, then the above result could be said that any $$\Delta $$Δ-critical graph with n vertices that satisfies $$\Delta \ge \frac{n}{2}$$Δ≥n2 is Hamiltonian. Since the overfull subgraph conjecture is still open, the natural question is how to directly prove a $$\Delta $$Δ-critical graph with n vertices that satisfies $$\Delta \ge \frac{n}{2}$$Δ≥n2 is Hamiltonian. Luo and Zhao (J Graph Theory 73:469–482, 2013) show that a $$\Delta $$Δ-critical graph with n vertices that satisfies $$\Delta \ge \frac{6n}{7}$$Δ≥6n7 is Hamiltonian. In this paper, by developing new lemmas for critical graphs, we show that if G is a $$\Delta $$Δ-critical graph with n vertices satisfying $$\Delta \ge \frac{4n}{5}$$Δ≥4n5, then G is Hamiltonian.