We study graphs that are simultaneously regular with respect to the ordinary vertex degree and regular with respect to the triangle degree, that is, the number of triangles containing a given vertex. We call such graphs regular $K_3$-regular. We investigate the (non-)existence of regular $K_3$-regular graphs with prescribed parameters $(r_2,r_3)$, where $r_2$ is the vertex degree and $r_3$ is the triangle degree. General bounds relating vertex and edge triangle degrees are derived, and non-existence results are established for broad ranges of these parameters. Special attention is paid to Turán graphs, for which we establish uniqueness results for certain parameters. The paper concludes with a summary of admissible parameters and several open problems.

翻译：本文研究同时满足通常顶点度正则性和三角形度（即包含给定顶点的三角形数量）正则性的图。我们将此类图称为正则$K_3$-正则图。我们探究具有指定参数$(r_2,r_3)$的正则$K_3$-正则图的存在/不存在性，其中$r_2$为顶点度，$r_3$为三角形度。推导了顶点度与边三角形度之间的一般约束关系，并针对广范围的参数建立了不存在性结论。特别关注图兰图，对其特定参数证明了唯一性结论。本文最后总结了可允许参数集并提出了若干未解决问题。