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Computational methods for finding bi-regular cages

Jan Goedgebeur, Jorik Jooken, Tibo Van den Eede

TL;DR

An exhaustive generation algorithm is presented, which leads to previously unknown $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(r,m,g)$-cages for 24 different triples, and a theorem is generalized, leading to 73 additional improved upper bounds.

Abstract

An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.

Computational methods for finding bi-regular cages

TL;DR

An exhaustive generation algorithm is presented, which leads to previously unknown previously unknown exhaustive lists of -cages for 24 different triples, and a theorem is generalized, leading to 73 additional improved upper bounds.

Abstract

An -graph is a (simple, undirected) graph of girth with vertices of degrees and where . Given , we seek the -graphs of minimum order, called -cages or bi-regular cages, whose order is denoted by . In this paper, we use computational methods for finding -graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to previously unknown exhaustive lists of -cages for 24 different triples . This also leads to the improvement of the lower bound of from 66 to 69. Secondly, we improve 49 upper bounds of based on constructions that start from -regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.

Paper Structure

This paper contains 18 sections, 4 theorems, 9 equations, 3 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notation and definitions
  4. Exhaustive generation of $(\{r,m\};g)$-graphs
  5. Refined lower bound for $n(\{r,m\};g)$
  6. Pruning rule: distance between vertices of degree $m$
  7. Pruning rule: amount of vertices of degree $m$ in a BMT
  8. The backtracking algorithm
  9. Experiments and results
  10. Exhaustive lists
  11. Improved lower bounds
  12. Constructions for bi-regular graphs out of regular graphs
  13. Experiments and results
  14. Generalized upper bound by gluing $(r,g)$-graphs
  15. Experiments and results
  16. ...and 3 more sections

Key Result

Theorem 2.4

Suppose $T_{r,m,d}^{(\mathsf{w})}$ is a BMT, $3 \leq r < m$ and $d \geq 2$. Then

Figures (3)

  • Figure 1: A BMT $T_{3,4,3}^{(v_{\mathsf{w}})}$.
  • Figure 2: One of the $(\{3,7\};5)$-cages, which is also hypohamiltonian.
  • Figure 3: One of the $(\{3,4\};6)$-cages (left) and one of the $(\{4,5\};5)$-cages (right).

Theorems & Definitions (11)

  • Definition 2.1: Bi-regular Moore tree (BMT)
  • Example 2.2: Bi-regular Moore tree (BMT)
  • proof
  • Theorem 2.4
  • Theorem 2.5: biregEven and Aurajo-Pardo2007
  • Definition 2.6: $m$-placement
  • Example 2.7: $m$-placement
  • Definition 2.8: Maximal $m$-placement
  • Theorem 4.1: semiCubic2023
  • Theorem 4.2
  • ...and 1 more