Constructions of minimally $t$-tough regular graphs

Abstract: 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 $\tau(G)$ 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 $\tau(G)=t$ and $\tau(G-e)<t$ 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 $3k+1~(k\geq 5)$.