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New small regular graphs of given girth: the cage problem and beyond

Geoffrey Exoo, Jan Goedgebeur, Jorik Jooken, Louis Stubbe, Tibo Van den Eede

TL;DR

This work tackles the cage problem by developing a layered, lift-based computational framework to construct $(k,g)$-graphs of minimal order and by extending these methods to several variants. The authors implement four complementary algorithms—exhaustive lifts, a tabu-search heuristic, hill climbing, and excision—to generate small regular graphs with prescribed girth, achieving eleven new upper bounds for the classical cage problem and substantial improvements for several variants. They provide rigorous validation, public release of code and graphs, and demonstrate the approach's potential to discover new spectra and building blocks for future constructions. The results underscore the effectiveness of voltage-graph lifts for extremal graph theory problems and offer practical pathways toward discovering infinite families of cages.

Abstract

The cage problem concerns finding $(k,g)$-graphs, which are $k$-regular graphs with girth $g$, of the smallest possible number of vertices. The central goal is to determine $n(k,g)$, the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: $n(3,16) \leq 936$, $n(3,17) \leq 2048$, $n(4,9) \leq 270$, $n(4,10) \leq 320$, $n(4,11) \leq 713$, $n(5,9) \leq 1116$, $n(6,11) \leq 7783$, $n(8,7) \leq 774$, $n(10,7) \leq 1608$, $n(12,7) \leq 2890$ and $n(14,7) \leq 4716$. Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for $n(4,10)$, from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.

New small regular graphs of given girth: the cage problem and beyond

TL;DR

This work tackles the cage problem by developing a layered, lift-based computational framework to construct -graphs of minimal order and by extending these methods to several variants. The authors implement four complementary algorithms—exhaustive lifts, a tabu-search heuristic, hill climbing, and excision—to generate small regular graphs with prescribed girth, achieving eleven new upper bounds for the classical cage problem and substantial improvements for several variants. They provide rigorous validation, public release of code and graphs, and demonstrate the approach's potential to discover new spectra and building blocks for future constructions. The results underscore the effectiveness of voltage-graph lifts for extremal graph theory problems and offer practical pathways toward discovering infinite families of cages.

Abstract

The cage problem concerns finding -graphs, which are -regular graphs with girth , of the smallest possible number of vertices. The central goal is to determine , the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: , , , , , , , , , and . Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for , from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.

Paper Structure

This paper contains 19 sections, 2 theorems, 16 equations, 4 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of lifts
  4. Algorithms for generating regular graphs of given girth
  5. Selection and generation of groups and base graphs
  6. Selection of voltage assignments
  7. Structure-based exclusions
  8. Assignment-based exclusions
  9. Efficient filtering
  10. Iterating over voltage assignments
  11. Exhaustive backtracking
  12. Heuristic search
  13. Hyperparameters
  14. Hill climbing heuristic
  15. Excision
  16. ...and 4 more sections

Key Result

Proposition 1

Given a graph $G$, a group $\Gamma$ and a voltage assignment $\alpha$ with corresponding lift $G^\alpha$. The girth of $G^\alpha$ is equal to the length of the shortest, closed, non-reversing walk with net voltage $0_\Gamma$ in the digraph associated with $G$.

Figures (4)

  • Figure 1: A lift of the dumbbell graph with the group $C_5$ resulting in the Petersen graph. Opposing arcs in the lift were replaced with a single edge.
  • Figure 2: The resulting $3$-regular graphs with parallel edges, loops and semi-edges obtained through multigraph+. Only degree distributions yielding graphs are shown. The graphs marked in red are isomorphic.
  • Figure 3: The excised set of vertices indicated in red from the $(4,12)$-cage (left) and the $(6,12)$-cage (right).
  • Figure 4: The base graphs and groups that resulted in a new upper bound for the cage problem. Darts corresponding to unmarked edges are assigned the neutral element. For Figs. \ref{['fig:cageRecord316']}, \ref{['fig:cageRecord317']} and \ref{['fig:cageRecord59']}, group elements are numbered by their index as given by GAP's MultiplicationTable.

Theorems & Definitions (2)

  • Proposition 1: exoo2011voltagegraphlifts
  • Proposition 2: topologicalGraphTheory