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Minimal Regular graphs with every edge in a triangle

James Preen

Abstract

Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then classify all such graphs using line graphs and even cycle decompositions. Examples of ways to create such r-regular graphs with r >= 6 are also given. In the 5-regular case, these minimal graphs are proven to be the only regular graphs with every edge in a triangle which cannot have an edge removed and still have every edge in a triangle.

Minimal Regular graphs with every edge in a triangle

Abstract

Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then classify all such graphs using line graphs and even cycle decompositions. Examples of ways to create such r-regular graphs with r >= 6 are also given. In the 5-regular case, these minimal graphs are proven to be the only regular graphs with every edge in a triangle which cannot have an edge removed and still have every edge in a triangle.

Paper Structure

This paper contains 8 sections, 9 theorems, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multiple edges
  3. Even regularity
  4. Minimal but not triminimal graphs
  5. Triminimal quintic graphs
  6. Construction of all triminimal quintic graphs
  7. Diamonds and even-cycle decompositions
  8. Removable edges

Key Result

Theorem 1.1

Suppose $G$ is an $r$-regular graph with the triangle property. Every $v\in V(G)$ is incident with at least $\frac{r}{2}$ triangles.

Figures (7)

  • Figure 1: A quintic multigraph with 12 vertices and 14 triangles
  • Figure 2: Edges in Petersen at distance 3 giving a 6 regular graph with 15 triangles
  • Figure 3: Replacing a triangle by two new vertices
  • Figure 4: The unique quintic graph $L_7$ containing 14 vertices and 14 triangles
  • Figure 5: Double cover of $E(L_7)$ by a sequence of diamonds
  • ...and 2 more figures

Theorems & Definitions (18)

  • Theorem 1.1
  • proof
  • Corollary 1.1
  • proof
  • Theorem 1.2
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 8 more