Regular $K_3$-irregular graphs

TL;DR

This paper investigates the existence of regular graphs that are $K_3$-irregular, establishing bounds, providing examples, and developing algorithms to find such graphs for various degrees.

Contribution

It proves non-existence for regularities up to 7, narrows bounds for regularity 8, provides an explicit 9-regular example, and introduces an evolutionary algorithm for finding more such graphs.

Findings

No regular $K_3$-irregular graphs with degree at most 7.

Explicit 9-regular $K_3$-irregular graph constructed.

Algorithm developed to discover graphs with degrees 9 to 30.

Abstract

We address the problem proposed by Chartrand, Erd\H{o}s and Oellermann (1988) about the existence of regular $K_3$-irregular graphs. We first establish bounds on the $K_3$-degrees of such graphs and use them to prove that there are no such graphs with regularities at most $7$. For the regularity $8$, we narrow down the bounds on the order of such graphs to six possible values. We then present an explicit example of a $9$-regular $K_3$-irregular graph. Finally, we discuss an evolutionary algorithm developed to discover more examples of $r$-regular $K_3$-irregular graphs for consecutive values $r \in \{9, \dots, 30\}$.