Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonexistence of certain edge-girth-regular graphs

Leen Droogendijk

Abstract

Edge-girth-regular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no $3$-regular \emph{egr} graph with girth $7$ such that every edge is on exactly $6$ shortest cycles, and there is no $3$-regular \emph{egr} graph with girth $8$ such that every edge is on exactly $14$ shortest cycles. This was conjectured by Goedgebeur and Jooken. A few other unresolved cases are settled as well.

Nonexistence of certain edge-girth-regular graphs

Abstract

Edge-girth-regular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no -regular \emph{egr} graph with girth such that every edge is on exactly shortest cycles, and there is no -regular \emph{egr} graph with girth such that every edge is on exactly shortest cycles. This was conjectured by Goedgebeur and Jooken. A few other unresolved cases are settled as well.
Paper Structure (9 sections, 25 theorems, 4 figures)

This paper contains 9 sections, 25 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Some notational conventions.
  3. A few lemmas.
  4. $egr(v,4,4,4)$-graphs.
  5. $egr(v,3,8,2t)$-graphs.
  6. $egr(v,3,8,14)$-graphs.
  7. $egr(v,k,g,(k-1)^{\frac{g-1}{2}}-1)$-graphs for $g$ odd and $g\geq5$.
  8. $egr(v,3,7,6)$-graphs.
  9. Discussion

Key Result

Lemma 1

Let $G$ be an $egr(v,k,g,\lambda)$-graph. Then every vertex of $G$ is contained in exactly $\frac{k\lambda}{2}$ shortest cycles.

Figures (4)

  • Figure 1: Developing $D_2$ for an $egr(v,4,4,4)$-graph, case 1
  • Figure 2: Developing $D_2$ for an $egr(v,4,4,4)$-graph, case 2
  • Figure 3: A spanning tree for $D_3$
  • Figure 4: A spanning tree for $D$

Theorems & Definitions (50)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 40 more