Hamiltonian cycles in tough $(P_4 \cup P_1)$-free graphs
Songling Shan
TL;DR
This work proves that every 23-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian, addressing a question posed by Broersma regarding Chvátal's toughness conjecture in this restricted graph class. The authors develop a multi-layer strategy: identify a cutset $S$ so that $G-S$ is $P_4$-free, build a cycle that covers all of $G-S$, and then iteratively insert the $S$-vertices using large neighborhood and matching arguments. Key ingredients include generalized $K_{1,r}$-matchings, cycle-cover decompositions, König's theorem, and the Häggkvist–Thomassen result on cycles through prescribed edges, all tailored to the $(P_4\cup P_1)$-free structure. The approach yields a constructive Hamiltonian cycle under a relatively large toughness bound and advances the understanding of Chvátal's conjecture within this important graph class.
Abstract
In 1973, Chvátal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes of graphs. Notably, Jung in 1978 proved that every 1-tough $P_4$-free graph on at least three vertices is Hamiltonian. However, the problem remains challenging even when restricted to graphs with no induced $P_4\cup P_1$, the disjoint union of a path on four vertices and a one-vertex path. In 2013, Nikoghosyan conjectured that every 1-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. Later in 2015, Broersma remarked that ``this question seems to be very hard to answer, even if we impose a higher toughness." He instead posed the following question: ``Is the general conjecture of Chvátal's true for $(P_4\cup P_1)$-free graphs?" We provide a positive answer to Broersma's question by establishing that every $23$-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian.