Hamiltonian cycles in 7-tough $(P_4\cup P_1)$-free graphs

Abstract

Shan~[J. Graph Theory (2026)] proved that every 23-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. We improve this bound to 7 by replacing the final cut analysis in Shan's framework with an asymmetric separation criterion and a cograph covering lemma.