On the Toughness of Regular Graphs and Prisms

TL;DR

This work investigates the toughness of regular graphs and their prisms, focusing on when $\tau(G)$ attains the theoretical maximum $\frac{r}{2}$ (supertough) and how prism operations affect toughness. It proves a nonexistence result for a $5$-regular supertough graph of order $18$ via exhaustive search, and constructs an infinite family of $4$-regular graphs with toughness $2$ that contain claws. The authors derive general lower and upper bounds for the toughness of prisms, establish when a prism preserves or increases toughness, and show that for inflations of degree $r$, prisms can achieve toughness $\frac{r}{2}$ under suitable conditions. Additional results cover upper bounds on $\tau(G^p)$, including the precise behavior for graphs with small toughness and for cubic and bipartite cases. Altogether, the paper deepens understanding of how regularity, inflation, and prism operations interact with graph toughness and Hamiltonicity implications.

Abstract

We contribute results on $r$-regular graphs that do and don't have the maximum possible toughness, namely $r/2$. Doty and Ferland showed the existence of a $5$-regular graph with toughness $5/2$ for all even orders except $n= 18$. Using a computer search we show that there does not exist such a graph for $n=18$. Also, we provide the first family of $4$-regular graphs with toughness $2$ that contains claws. For the prism $G \Box K_2$ of a graph~$G$, we provide several bounds including a sufficient condition for the prism to have the same toughness as~$G$. In particular, we show that if $G$ has toughness $t\le \frac{1}{2}$ then its prism has toughness $2t$; further, the prism of any $r$-regular $r$-connected inflation has toughness~$r/2$ (despite being $(r+1)$-regular) and in general the prism of any $3$-regular graph has toughness at most~$3/2$.

Theorems & Definitions (14)

• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10