Constructions of minimally $t$-tough regular graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915068857286656 |
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| author | Cheng, Kun Li, Chengli Liu, Feng |
| author_facet | Cheng, Kun Li, Chengli Liu, Feng |
| contents | A non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The toughness $τ(G)$ of the graph $G$ is the maximum value of $t$ such that $G$ is $t$-tough. A graph $G$ is said to be minimally $t$-tough if $τ(G)=t$ and $τ(G-e)<t$ for every $e\in E(G)$. In 2003, Kriesell conjectured that every minimally $1$-tough graph contains a vertex of degree $2$. In 2018, Katona and Varga generalized this conjecture, asserting that every minimally $t$-tough graph contains a vertex of degree $\lceil 2t \rceil$. Recently, Zheng and Sun disproved the generalized Kriesell conjecture by constructing a family of $4$-regular graphs of even order. They also raised the question of whether there exist other minimally $t$-tough regular graphs that do not satisfy the generalized Kriesell conjecture. In this paper, we provide an affirmative answer by constructing a family of $4$-regular graphs of odd order, as well as a family of 6-regular graphs of order $3k+1~(k\geq 5)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructions of minimally $t$-tough regular graphs Cheng, Kun Li, Chengli Liu, Feng Combinatorics A non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The toughness $τ(G)$ of the graph $G$ is the maximum value of $t$ such that $G$ is $t$-tough. A graph $G$ is said to be minimally $t$-tough if $τ(G)=t$ and $τ(G-e)<t$ for every $e\in E(G)$. In 2003, Kriesell conjectured that every minimally $1$-tough graph contains a vertex of degree $2$. In 2018, Katona and Varga generalized this conjecture, asserting that every minimally $t$-tough graph contains a vertex of degree $\lceil 2t \rceil$. Recently, Zheng and Sun disproved the generalized Kriesell conjecture by constructing a family of $4$-regular graphs of even order. They also raised the question of whether there exist other minimally $t$-tough regular graphs that do not satisfy the generalized Kriesell conjecture. In this paper, we provide an affirmative answer by constructing a family of $4$-regular graphs of odd order, as well as a family of 6-regular graphs of order $3k+1~(k\geq 5)$. |
| title | Constructions of minimally $t$-tough regular graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.12659 |