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On extremal (almost) edge-girth-regular graphs

Gabriela Araujo-Pardo, György Kiss, István Porupsánszki

Abstract

A $k$-regular graph of girth $g$ is called edge-girth-regular graph, shortly egr-graph, if each of its edges is contained in exactly $λ$ distinct $g-$cycles. An egr-graph is called extremal for the triple $(k, g, λ)$ if has the smallest possible order. We prove that some graphs arising from incidence graphs of finite planes are extremal egr-graphs. We also prove new lower bounds on the order of egr-graphs.

On extremal (almost) edge-girth-regular graphs

Abstract

A -regular graph of girth is called edge-girth-regular graph, shortly egr-graph, if each of its edges is contained in exactly distinct cycles. An egr-graph is called extremal for the triple if has the smallest possible order. We prove that some graphs arising from incidence graphs of finite planes are extremal egr-graphs. We also prove new lower bounds on the order of egr-graphs.
Paper Structure (4 sections, 21 theorems, 68 equations, 3 figures)

This paper contains 4 sections, 21 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Edge-girth-regular graphs arising from finite geometries
  3. Almost edge-girth-regular graphs
  4. Lower bounds on the order of girth-regular graphs

Key Result

Theorem 1.1

Let $G$ be a $(k,g)$-graph with $k>2$. Then the order of $G$ is at least $n_0(k,g)$, where

Figures (3)

  • Figure 1: Deleted points, Theorem 2.3.
  • Figure 2: Deleted points, Theorem 2.4.
  • Figure 3: Deleted points, Theorem 2.5.

Theorems & Definitions (48)

  • Theorem 1.1: Moore bound
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Drglin, Filipovski, Jajcay, Raiman
  • Theorem 1.5: Porupsánszki
  • Definition 2.1
  • Theorem 2.2: Gács, Héger
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 38 more