Regular $K_3$-regular graphs

TL;DR

This paper explores the existence and properties of graphs that are simultaneously regular in vertex degree and triangle degree, providing bounds, non-existence results, and characterizations for special cases like Turán graphs.

Contribution

It introduces the concept of regular $K_3$-regular graphs, derives bounds, proves non-existence for certain parameters, and characterizes Turán graphs within this framework.

Findings

Established bounds relating vertex and triangle degrees.

Proved non-existence of regular $K_3$-regular graphs for broad parameter ranges.

Proved uniqueness of Turán graphs for specific parameters.

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.