Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected

Feng Liu

Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected

Combinatorics 2025-03-18 v1

Authors: Feng Liu

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.

Keywords

graph theory Hamiltonian-connected toughness induced subgraph connectivity

Cite

@article{arxiv.2503.12860,
  title  = {Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected},
  author = {Feng Liu},
  journal= {arXiv preprint arXiv:2503.12860},
  year   = {2025}
}

Every $2k$-connected $(P_2\cup kP_1)$-free graph with toughness greater than one is hamiltonian-connected

Abstract

Keywords

Cite

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