Regular $K_3$-regular graphs
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912970449092608 |
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| author | Hak, Artem Kozerenko, Sergiy Lohvynov, Denys Yarosh, Yurii |
| author_facet | Hak, Artem Kozerenko, Sergiy Lohvynov, Denys Yarosh, Yurii |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23517 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regular $K_3$-regular graphs Hak, Artem Kozerenko, Sergiy Lohvynov, Denys Yarosh, Yurii Combinatorics Discrete Mathematics 05C07, 05C35, 05C99 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. |
| title | Regular $K_3$-regular graphs |
| topic | Combinatorics Discrete Mathematics 05C07, 05C35, 05C99 |
| url | https://arxiv.org/abs/2602.23517 |