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arXiv · 2503.06466

Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs

L. C. Eze·R. Jajcay·T. Jajcayová·D. Závacká
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Abstract

The Cage Problem requires for a given pair $k \geq 3, g \geq 3$ of integers the determination of the order of a smallest $k$-regular graph of girth $g$. We address a more general version of this problem and look for the $(k,g)$-spectrum of orders of $(k,g)$-graphs: the (infinite) list of all orders of $(k,g)$-graphs. By establishing these spectra we aim to gain a better understanding of the structure and properties of $(k,g)$-graphs and hope to use the acquired knowledge in both determining new orders of smallest $k$-regular graphs of girth $g$ as well as developing a set of tools suitable for constructions of extremal graphs with additional requirements. We combine theoretical results with computer-based searches, and determine or determine up to a finite list of unresolved cases the $(k,g)$-spectra for parameter pairs for which the orders of the corresponding cages have already been established.

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L. C. Eze, R. Jajcay, T. Jajcayová, D. Závacká. 2025-03-09. Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs. https://arxiv.org/abs/2503.06466

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