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arXiv · 2107.08476

Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs

Yuping Gao·Songling Shan
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Abstract

The toughness of a noncomplete graph $G$ is the maximum real number $t$ such that the ratio of $|S|$ to the number of components of $G-S$ is at least $t$ for every cutset $S$ of $G$, and the toughness of a complete graph is defined to be $\infty$. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.

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Yuping Gao, Songling Shan. 2021-07-18. Hamiltonian cycles in 7-tough $(P_3\cup 2P_1)$-free graphs. https://arxiv.org/abs/2107.08476

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