Regular $K_3$-regular graphs

Artem Hak; Sergiy Kozerenko; Denys Lohvynov; Yurii Yarosh

Regular $K_3$-regular graphs

Combinatorics 2026-03-18 v2 Discrete Mathematics

Authors: Artem Hak , Sergiy Kozerenko , Denys Lohvynov , Yurii Yarosh

Abstract

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 Tur\'an graphs, for which we establish uniqueness results for certain parameters. The paper concludes with a summary of admissible parameters and several open problems.

Keywords

turán theory graph theory triangulations

Cite

@article{arxiv.2602.23517,
  title  = {Regular $K_3$-regular graphs},
  author = {Artem Hak and Sergiy Kozerenko and Denys Lohvynov and Yurii Yarosh},
  journal= {arXiv preprint arXiv:2602.23517},
  year   = {2026}
}

Comments

Second 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 Tur\'an graphs with new results. Any comments are welcome

Regular $K_3$-regular graphs

Abstract

Keywords

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