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arXiv · 2311.08634

On the minimum degree of minimally $ t $-tough, claw-free graphs

Hui Ma·Xiaomin Hu·Weihua Yang
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Abstract

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil $ and proved that the minimum degree of minimally $ \frac{1}2 $-tough and $ 1 $-tough, claw-free graphs is 1 and 2, respectively. We have show that every minimally $ 3/2 $-tough, claw-free graph has a vertex of degree of $ 3 $. In this paper, we give an upper bound on the minimum degree of minimally $t$-tough, claw-free graphs for $ t\geq 2 $.

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Hui Ma, Xiaomin Hu, Weihua Yang. 2023-11-15. On the minimum degree of minimally $ t $-tough, claw-free graphs. https://arxiv.org/abs/2311.08634

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