Regular $K_3$-Irregular Graphs of Every Regularity at Least Nine

Abstract

For a vertex $v$ of a graph $G$, the triangle-degree $\operatorname{td}_G(v)$ is the number of triangles containing $v$. A graph is triangle-distinct, or $K_3$-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erdős, and Oellermann asked whether a regular $K_3$-irregular graph exists. We prove that such graphs exist for every regularity $r\ge 9$. More precisely, for every integer $k\ge 15$ we construct a $2k$-regular triangle-distinct graph on $4k+2$ vertices; complementation gives a $(2k+1)$-regular example of the same order. The construction uses two antiregular threshold blocks joined by a zero-one matrix with prescribed margins, followed by matrix $2$-switches that preserve those margins. After a uniform reference perturbation, exactly three triangle-degree collisions remain. Switches chosen according to parity remove two of them, and the last possible collision is controlled by a short quadratic discriminant argument. Odd $k\ge 17$ and even $k\ge 62$ are handled symbolically, while the remaining 24 values are settled by exact finite verification. Together with the known examples for $9\le r\le 29$, this closes the positive existence problem for every $r\ge 9$.