On the order-diameter ratio of girth-diameter cages

2025-11-26

AI Summary

This work investigates the asymptotic ratio of the order $n(k;g,d)$ of $(k;g,d)$-cages—minimum $k$-regular graphs with girth $g$ and diameter $d$—to the diameter $d$, for fixed $k$ and $g$. Method: Theoretically, we establish tight asymptotic upper and lower bounds on $n(k;g,d)/d$ as $d o infty$, and prove its computability in constant time. Algorithmically, we design and implement an exhaustive generation framework leveraging constraint satisfaction and symmetry-breaking pruning. Contribution/Results: We resolve long-standing open cases by exactly determining $n(3;4,d)$ and $n(3;5,d)$, deriving their closed-form expressions. We construct and verify several previously unknown cages, including a $(3;7,35)$-cage of order 136, and provide the most comprehensive constructive catalogue and enumeration results to date.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• 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: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$. We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g in {4,5}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136.

Problem

• Study asymptotic bounds for order-diameter ratio in girth-diameter cages
• Determine exact values and counts of cages for specific girth parameters
• Design algorithm to generate and classify cages for open cases

Innovation

• Established asymptotic bounds for order-diameter ratio
• Developed constant-time computation for fixed parameters
• Designed exhaustive graph generation algorithm for cages