Papers › Exhaustive generation of edge-girth-regular graphs
Exhaustive generation of edge-girth-regular graphs
Jan Goedgebeur, Jorik Jooken
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Edge-girth-regular graphs (abbreviated as egr graphs) are a class of highly 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. We use this as one of the building blocks to propose another algorithm that can exhaustively generate all egr(v,k,g,λ) graphs for fixed parameters v, k, g and λ. We implement this algorithm and use it in a large-scale computation to obtain several new extremal graphs and improvements for lower and upper bounds from the literature 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)=∞.
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