Improved lower bounds on the maximum size of graphs with girth 5

2025-08-07

AI Summary

This study addresses the problem of improving the lower bound on the extremal function $ ext{ex}(n; {C_3, C_4}) $—i.e., the maximum number of edges in an $ n $-vertex graph of girth at least 5. To overcome the limitation of conventional hill-climbing heuristics, which often stagnate at local optima, we propose a structure-aware multi-round optimization framework. It initializes the search with near-extremal configurations from graphs of comparable size and employs iterative propagation to transfer high-quality substructural patterns across different graph orders. Systematically applied for $ n in [74, 198] $, our method improves all previously known lower bounds for $ ext{ex}(n; {C_3, C_4}) $, except at $ n = 96 $ and $ n = 97 $. This approach substantially enhances the global exploration capability and boundary-pushing efficacy of constructive (non-exact) methods, establishing a novel paradigm for density estimation under girth constraints in extremal graph theory.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

We present a new algorithm for improving lower bounds on $ex(n;{C_3,C_4})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by Exoo, McKay, Myrvold and Nadon (2011) to find small cages. Our algorithm considers a range of values of $n$ in multiple passes. In each pass, the hill-climbing heuristic for a specific value of $n$ is initialized with a few graphs obtained by modifying near-extremal graphs previously found for neighboring values of $n$, allowing to `propagate' good patterns that were found. Focusing on the range $nin {74,75, dots, 198}$, which is currently beyond the scope of exact methods, our approach yields improvements on existing lower bounds for $ex(n;{C_3,C_4})$ for all $n$ in the range, except for two values of $n$ ($n=96,97$).

Problem

• Improve lower bounds for maximum size of girth-5 graphs
• Develop algorithm to enhance ex(n;{C3,C4}) calculations
• Focus on n in range 74-198 for better bounds

Innovation

• Hill-climbing heuristic variant for graph bounds
• Multi-pass approach for neighboring value propagation
• Focus on range beyond exact methods scope