2602.23517Feb 26, 2026Second version. We added references to related works on (r,c)-constant graphs and vertex-girth-regular graphs, clarified the relation between our results and existing literature, and extended the section on Turán graphs with new results. Any comments are welcome
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 K3-regular. We investigate the (non-)existence of regular K3-regular graphs with prescribed parameters (r2,r3)
Nearby in the stack
, where
r2
is the vertex degree and
r3
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.