On minimally 1-tough $(K_1\cup P_4)$-free graphs

Shiyu Cao; Jing Chen; Wei Zheng

On minimally 1-tough $(K_1\cup P_4)$-free graphs

Combinatorics 2026-07-08 v1

Authors: Shiyu Cao , Jing Chen , Wei Zheng

Abstract

Agraph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It is conjectured that every $(K_1\cup P_4)$ -free 1-tough graph is hamiltonian. In this paper, we characterize the structure of minimally 1-tough $(K_1\cup P_4)$ -free graphs, and thus show that the above conjecture is true for minimally 1-tough graphs. Furthermore, it is also proved that the Kriesell's conjecture which states that each minimally 1-tough graph has a vertex of degree 2 holds for minimally 1-tough $(K_1\cup P_4)$ -free graphs.

Keywords

minimally 1-tough Hamiltonian forbidden induced subgraph graph toughness Kriesell's conjecture

Cite

@article{arxiv.2607.07239,
  title  = {On minimally 1-tough $(K_1\cup P_4)$-free graphs},
  author = {Shiyu Cao and Jing Chen and Wei Zheng},
  journal= {arXiv preprint arXiv:2607.07239},
  year   = {2026}
}