Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regular $K_3$-regular graphs

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. Furthermore, it is shown that the class of regular $K_3$-regular graphs is closed under the Cartesian product, and decomposability properties are analysed. 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.

Regular $K_3$-regular graphs

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 -regular. We investigate the (non-)existence of regular -regular graphs with prescribed parameters , where is the vertex degree and 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. Furthermore, it is shown that the class of regular -regular graphs is closed under the Cartesian product, and decomposability properties are analysed. 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.
Paper Structure (8 sections, 12 theorems, 57 equations, 2 figures, 1 table)

This paper contains 8 sections, 12 theorems, 57 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main definitions
  3. Establishing bounds
  4. Edge degree statements
  5. Non-trivial bounds on the parameters
  6. Cartesian product
  7. Turán graphs
  8. Open questions

Key Result

Lemma 3.1

For every graph with parameters $(r_2, r_3)$, the following upper bound holds:

Figures (2)

  • Figure 1: Constructions in Theorem \ref{['Pr:noDown']}.
  • Figure 2: The graphs $G_2$, $G_3$, $G_4$ and $G_5$ from Table \ref{['tab:admissible']}.

Theorems & Definitions (27)

  • Definition 2.1: Char-Erd-Oell:88Char:87
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 17 more