Regular $K_3$-irregular graphs

We address the problem proposed by Chartrand, Erdős 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 $r\in \{9,\dots ,30\}$.