On the Toughness of Regular Graphs and Prisms

TL;DR

This paper investigates the toughness of regular graphs and their prisms, providing new bounds, existence results, and specific constructions, including the first family of 4-regular graphs with toughness 2 containing claws.

Contribution

It presents new existence results for regular graphs with specific toughness, introduces the first family of 4-regular graphs with toughness 2 containing claws, and establishes bounds on the toughness of graph prisms.

Findings

No 5-regular graph with toughness 5/2 exists for n=18.

Constructed the first 4-regular graphs with toughness 2 containing claws.

Prism of a graph with toughness t ≤ 1/2 has toughness 2t.

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~$3/2$.