发表机构
School of International Trade and Economics, Central University of Finance and Economics（中央财经大学国际经济贸易学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明对每个正则度 $r\ge 9$ 存在正则 $K_3$-不规则图，通过构造 $2k$-正则图（$k\ge 15$）及补图得到 $(2k+1)$-正则图，利用阈值块和矩阵切换解决三角形度冲突，结合符号与有限验证，彻底回答了 Chartrand 等人的问题。
AI 中文摘要
对于图 $G$ 的顶点 $v$，三角形度 $\operatorname{td}_G(v)$ 是包含 $v$ 的三角形数量。如果图的顶点三角形度两两不同，则该图是三角形相异的，即 $K_3$-不规则的。Chartrand、Erdős 和 Oellermann 提出了是否存在正则 $K_3$-不规则图的问题。我们证明对于每个正则度 $r\ge 9$，这样的图都存在。更精确地，对于每个整数 $k\ge 15$，我们构造一个 $2k$-正则的三角形相异图，其有 $4k+2$ 个顶点；取补图则得到同阶的 $(2k+1)$-正则例子。该构造使用两个由具有规定边缘和零一矩阵连接的反正则阈值块，随后进行保持这些边缘的矩阵 $2$-切换。在统一参考扰动后，恰好剩下三个三角形度冲突。根据奇偶性选择的切换移除其中两个，最后一个可能的冲突由简短的二次判别式论证控制。奇数 $k\ge 17$ 和偶数 $k\ge 62$ 的情况以符号方式处理，而剩余的 24 个值通过精确的有限验证解决。连同已知的 $9\le r\le 29$ 的例子，这解决了所有 $r\ge 9$ 的正存在性问题。
英文摘要
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 $r\ge 9$.
Comments19 pages
