Andrii Serdiuk | Search arXiv

arXiv2026

There is no $8$-regular $K_3$-irregular graph

A graph is $K_3$-irregular if its vertices belong to pairwise distinct numbers of triangles. We prove that no $8$-regular $K_3$-irregular graph exists, settling the last unresolved case. Following the initial discovery of such graphs for regularities $r \in \{10,11,12\}$ (Stevanovi'c et al., 2024), our previous work (Hak et al., 2025) showed that no such graphs exist for $r \le 7$, provided the first example for $r=9$, and proved that any $8$-regular candidate must have between $17$ and $22$ vertices. We exclude these possible orders for $r=8$ by combining careful analysis of triangle degrees with integer linear programming techniques. Meanwhile, a recent construction (Zhang, 2026) established that regular $K_3$-irregular graphs do exist for all $r \ge 9$. Together with our results, this establishes that an $r$-regular $K_3$-irregular graph exists if and only if $r\geq 9$.

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arXiv2026

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 such graphs. Using it, we have found such graphs for consecutive regularities from $9$ to $30$.

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