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arXiv · 2607.07239

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

Shiyu Cao·Jing Chen·Wei Zheng
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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.

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Shiyu Cao, Jing Chen, Wei Zheng. 2026-07-08. On minimally 1-tough $(K_1\cup P_4)$-free graphs. https://arxiv.org/abs/2607.07239

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