Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected
Feng Liu
TL;DR
This paper proves that every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness $\tau(G)>1$ is hamiltonian-connected, thereby strengthening the link between toughness, connectivity, and Hamiltonian properties in a forbidden-induced-subgraph setting. The authors deploy a longest $(u,v)$-path framework and a sequence of intricate structural claims to show that the remainder of the graph outside the path must be highly constrained, ultimately contradicting $\tau(G)\!>\!1$ unless a Hamilton path exists between any pair of vertices. This result extends prior 1-tough and $2k$-connected cases related to Shi and Shan's conjecture, and emphasizes toughness > 1 as a sharp condition for hamiltonian-connectedness in this graph class. The findings have implications for the study of Hamiltonicity in forbidden-subgraph graphs and contribute to the broader understanding of how global graph properties interact with induced-subgraph constraints.
Abstract
Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been independently confirmed by Xu, Li, and Zhou, as well as by Ota and Sanka. Inspired by this, we prove that every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected.