On the existence of minimally tough graphs having large minimum degrees

TL;DR

The paper tackles the relationship between minimum degree and toughness in minimally $t$-tough graphs, focusing on generalized conjectures that predicted small minimum degrees. It disproves the generalized Kriesell-type conjecture in planar graphs and their line graphs by constructing infinite families of minimally $t$-tough claw-free graphs with minimum degree approaching $3t$, including a $4$-regular planar counterexample with $t=\tfrac{3}{2}$ and a planar $7$-regular example arising as a square of a planar line graph. It further provides non-regular and regular claw-free constructions, showing that $\delta(G)$ can closely approach $3t$ and that the ratio $\delta(G)/t(G)$ can be made arbitrarily large. Motivated by these findings, the authors propose a revised conjecture denying a universal linear bound $\delta(G)\le \lceil c t\rceil$, and they supply evidence from $2$-solid and circulant-constructed graphs, along with open questions on the existence of any fixed bound $f(t)$.

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 $\lceil c t \rceil$.

Figures (6)

• Figure 1: All minimally $t(G)$-tough graphs $G$ of order at most $11$ satisfying $\delta(G) > \lceil 2t(G)\rceil$, where $t(G)\in \{4/3, 3/2, 6/4, 5/2\}$, respectively (from left to right).
• Figure 2: A minimally $3/2$-tough planar $4$-regular graph of order $24$.
• Figure 3: The square of the planar graph $H$ is a minimally $3$-tough $7$-regular graph.
• Figure 4: The line graph of the left graph is a minimally $5/2$-tough graph $G$ satisfying $\delta(G)=6$ (which is shown in the right).
• Figure 5: Two minimally $2$-tough graphs with minimum degree $5$ (regular and non-regular).