On hamiltonicity of 1-tough triangle-free graphs

Wei Zheng, Hajo Broersma, Ligong Wang


Let ω(G) denote the number of components of a graph G. A connected graph G is said to be 1-tough if ω(G − X)≤|X| for all X ⊆ V(G) with ω(G − X)>1. It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in general, and even not for triangle-free graphs. We present two classes of triangle-free graphs for which the reverse statement holds, i.e., for which hamiltonicity and 1-toughness are equivalent. Our two main results give partial answers to two conjectures due to Nikoghosyan.


toughness; 1-tough; forbidden subgraph; hamiltonicity; triangle-free graph

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DOI: http://dx.doi.org/10.5614/ejgta.2021.9.2.15


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