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

L. C. Eze, R. Jajcay, T. Jajcayová, D. Závacká

TL;DR

This work broadens the Cage Problem by targeting the full $(k,g)$-spectrum, the set of all orders of connected $k$-regular graphs of girth $g$, and develops a framework of theoretical results and computational tools to determine or bound these spectra. It establishes key structural properties, including closure of the spectrum under addition and positive multiples, and leverages diverse constructions (edge/vertex deletions, Moore-tree deletions, circulant and Petersen-type graphs, and voltage-based double covers) to generate large families of $(k,g)$-graphs from cages. The authors provide comprehensive results for several parameter pairs with known cages, delivering completely determined spectra (notably for many $(3,g)$, $(4,g)$, and $(5,g)$) and documenting numerous incomplete cases with ongoing work and bounds on $N(k,g)$. The study offers a practical, computationally driven methodology that complements theory, with implications for constructing extremal graphs and advancing understanding of the cage problem, while outlining open questions such as the bipartiteness of even-girth cages and the existence of certain Moore-type graphs.

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.

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

TL;DR

This work broadens the Cage Problem by targeting the full -spectrum, the set of all orders of connected -regular graphs of girth , and develops a framework of theoretical results and computational tools to determine or bound these spectra. It establishes key structural properties, including closure of the spectrum under addition and positive multiples, and leverages diverse constructions (edge/vertex deletions, Moore-tree deletions, circulant and Petersen-type graphs, and voltage-based double covers) to generate large families of -graphs from cages. The authors provide comprehensive results for several parameter pairs with known cages, delivering completely determined spectra (notably for many , , and ) and documenting numerous incomplete cases with ongoing work and bounds on . The study offers a practical, computationally driven methodology that complements theory, with implications for constructing extremal graphs and advancing understanding of the cage problem, while outlining open questions such as the bipartiteness of even-girth cages and the existence of certain Moore-type graphs.

Abstract

The Cage Problem requires for a given pair of integers the determination of the order of a smallest -regular graph of girth . We address a more general version of this problem and look for the -spectrum of orders of -graphs: the (infinite) list of all orders of -graphs. By establishing these spectra we aim to gain a better understanding of the structure and properties of -graphs and hope to use the acquired knowledge in both determining new orders of smallest -regular graphs of girth 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 -spectra for parameter pairs for which the orders of the corresponding cages have already been established.

Paper Structure

This paper contains 21 sections, 7 theorems, 5 equations, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Basic Concepts and Definitions
  3. Properties of (k,g)-Spectra
  4. (k,g)-Spectra for Parameters with Known n(k,g)
  5. Completely Determined (3,g)-Spectra
  6. Incomplete (3,g)-Spectra
  7. Completely Determined (4,g)-Spectra
  8. Incomplete (4,g)-Spectra
  9. Completely Determined (5,g)-Spectra
  10. Incomplete (5,g)-Spectra
  11. Other Completely Determined (k,g)-Spectra
  12. Computational Techniques
  13. Edge Subdivision Constructions
  14. Edge Deletion and Vertex Addition Constructions
  15. Vertex Deletion Constructions
  16. ...and 6 more sections

Key Result

Lemma 1

Assume that $k \geqslant 3$, $g \geqslant 3$, and let $\Gamma_1, \Gamma_2$ be two $(k,g)$-graphs of orders $n_1$ and $n_2$, respectively, with the additional property that at least one of the two graphs contains an edge not contained in a $g$-cycle or that it contains at least two distinct $g$-cycle

Theorems & Definitions (9)

  • Lemma 1: eze2023algorithmic
  • Corollary 2
  • Theorem 3: eze2023algorithmic
  • Theorem 4
  • proof
  • Theorem 5: BigIto
  • Corollary 6
  • Definition 1
  • Theorem 7: biggs1998constructions