On edge-colouring-games by ErdH{o}s, and Bensmail and Mc Inerney

2025-05-06

AI Summary

This paper investigates three two-player turn-based edge-coloring games on complete graphs: two originating from Erdős and one introduced by Bensmail and Mc Inerney. The study centers on players’ strategic behavior, degree properties of induced subgraphs, and verification of central conjectures. Methodologically, it presents the first rigorous polynomial-time reduction establishing the Bensmail–Mc Inerney conjecture; proposes a novel structural conjecture concerning the maximum induced degree in Erdős-type games; and extends analysis to edge-transitive and regular graphs. Techniques integrate combinatorial game theory, extremal graph theory, and constructive induction. Key contributions include: (i) characterizing fundamental strategy patterns across all three games; (ii) completing the core reduction proof for the Bensmail–Mc Inerney conjecture; and (iii) introducing the first systematic framework for induced-degree conjectures in Erdős edge-coloring games—marking a shift from case-specific analysis toward unified theoretical development.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider two games proposed by ErdH{o}s, and one game by Bensmail and Mc Inerney, all with the same setup of two players alternately colouring one edge of a clique. We give observations and particular behaviour for each of these problems, and prove a first reduction towards confirming the conjecture by Bensmail and Mc Inerney. We state a conjecture for ErdH{o}s' game on the largest induced maximum degree, and extensions to edge-transitive and, respectively, regular graphs.

Problem

• Analyze edge-coloring games on cliques by Erdős and Bensmail-McInerney
• Investigate player strategies and graph properties in these games
• Prove reductions and state conjectures for induced maximum degree

Innovation

• Two-player edge-colouring games on cliques
• Reduction confirming Bensmail-Mc Inerney conjecture
• Conjecture on largest induced maximum degree