On the existence of minimally tough graphs having large minimum degrees

Abstract

Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with degree precisely $\lceil 2t\rceil$, where $t$ is a positive real number. This conjecture has been recently verified for several families of graphs. For example, Ma, Hu, and Yang (2023) confirmed it for claw-free minimally $3/2$-tough graphs. Recently, Zheng and Sun (2024) disproved this conjecture by constructing a family of $4$-regular graphs with toughness approaching to $1$. In this paper, we disprove this conjecture for planar graphs and their line graphs. In particular, we construct an infinite family of minimally $t$-tough non-regular claw-free graphs with minimum degree close to thrice their toughness. This construction not only disproves a renewed version of Generalized Kriesel's Conjecture on non-regular graphs proposed by Zheng and Sun (2024), it also gives a supplement to a result due to Ma, Hu, and Yang (2023) who proved that every minimally $t$-tough claw-free graph with $t\ge 2$ has a vertex of degree at most $3t+ \lceil (t-5)/3\rceil$. Moreover, we conjecture that there is not a fixed constant $c$ such that every minimally $t$-tough graph has minimum degree at most $\lceil c t \rceil$.