Year: 2025
Type: article
Abstract: Edge-girth-regular graphs (abbreviated as egr graphs) are a class of regular graphs. More specifically, for integers v, k, g, and λ an egr(v,k,g,λ) graph is a k-regular graph with girth g on v vertices such that every edge is contained in exactly λ cycles of length g. The central problem in this paper is determining n(k,g,λ), which is defined as the smallest integer v such that an egr(v,k,g,λ) graph exists (or ∞ if no such graph exists) as well as determining the corresponding extremal graphs. We propose a linear time algorithm for computing how often an edge is contained in a cycle of length g, given a graph with girth g. Based on this we propose another algorithm that can exhaustively generate all egr(v,k,g,λ) graphs for fixed v, k, g, and λ. We use this algorithm in a large-scale computation to obtain several new extremal graphs and improvements for lower and upper bounds for n(k,g,λ). Among others, we show that n(3,6,2)=24,n(3,8,8)=40,n(3,9,6)=60,n(3,9,8)=60,n(4,5,1)=30,n(4,6,9)=35,n(6,5,20)=42 and we disprove a conjecture made by Araujo-Pardo and Leemans [Discrete Math.345(10):112991 (2022)] for the cubic girth 8 and girth 12 cases. Based on our computations, we conjecture that n(3,7,6)=n(3,8,10)=n(3,8,12)=n(3,8,14)=∞.
Source: Experimental Mathematics
Authors: Jan Goedgebeur, Jorik Jooken
Institutions: Ghent University, KU Leuven
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4410132900