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

2025-11-10

AI Summary

The cage problem seeks the minimum number $n(k,g)$ of vertices in a $k$-regular graph of girth $g$. This paper addresses this classical extremal graph theory problem—and several constrained variants—by introducing four novel computational construction methods: exhaustive lift-based enumeration, tabu search, a hybrid hill-climbing heuristic, and constraint-aware pruning techniques. These methods are systematically integrated into a unified framework adaptable to diverse structural constraints. Through large-scale, high-performance computational search, we improve the best-known upper bounds for 11 classical $(k,g)$ pairs; notably, $n(4,10)$ is reduced from 384 to 320, breaking a 22-year-old record. Our results substantially narrow the gaps between existing upper and lower bounds for multiple $n(k,g)$ values, advancing the state-of-the-art in computational cage construction and pushing the frontier of algorithmic solutions to the cage problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

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.

Problem

• Finding smallest regular graphs with given girth
• Developing computational methods for graph generation
• Establishing improved upper bounds for cage problem

Innovation

• Exhaustive lift generation for graph construction
• Tabu search heuristic for optimization process
• Hill climbing and excision techniques enhancement