Regular $K_3$-regular graphs
This addresses a theoretical problem in graph theory for researchers, focusing on incremental extensions of regularity concepts.
The paper tackles the problem of characterizing graphs that are simultaneously regular in vertex degree and triangle degree, called regular K3-regular graphs, by investigating existence and non-existence for prescribed parameters, with results including general bounds, non-existence proofs for broad parameter ranges, and uniqueness findings for Turán graphs.
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.