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arXiv · 2204.03630

Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

Elizabeth Grimm·Songling Shan·Anna Johnsen
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Abstract

For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that every $2K_2$-free $3/2$-tough graph with at least three vertices has a $2$-factor, and the toughness bound $3/2$ is best possible. In viewing $2K_2$, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest $R$ on 5, 6, or 7 vertices, we find the sharp toughness bound $t$ such that every $t$-tough $R$-free graph on at least three vertices has a 2-factor.

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Elizabeth Grimm, Songling Shan, Anna Johnsen. 2022-04-07. Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs. https://arxiv.org/abs/2204.03630

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