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On edge-colouring-games by Erdős, and Bensmail and Mc Inerney

Stijn Cambie, Michiel Provoost

TL;DR

This work advances the study of edge-colouring games on graphs by formulating a unified framework for three Erdős- and BM-inspired games, and analyzing both unbiased and biased variants. It combines strategy-stealing arguments, combinatorial game analysis, and computer-assisted enumeration to obtain new results: a first reduction toward Erdős’ and BM’s conjectures, a strategy-stealing derivation linking different clique sizes, and a complete resolution of the biased clique game at $(p,q)=(1,3)$ with explicit strategies. It also investigates behaviour on Colex graphs and irregular graphs to highlight the role of symmetry, and provides a detailed description of two independent computer implementations (C and Sage) that leverage Nauty for canonical labelling and backtracking to compute optimal outcomes. The results contribute to understanding when the second player can force wins in biased settings, extend known conjectures to new graph families, and provide a practical computational toolkit for exploring edge-colouring games on broader graph classes.

Abstract

We study two games proposed by Erdős, and one game by Bensmail and Mc Inerney, all sharing a common setup: two players alternately colour edges of a complete graph, or in the biased version, they colour $p$ and $q$ edges respectively on their turns, aiming to maximise a graph parameter determined by their respective induced subgraphs. In the unbiased case, we give a first reduction towards confirming the conjecture of Bensmail and Mc Inerney, propose a conjecture for Erdős' game on maximum degree, and extend the clique and maximum-degree versions to edge-transitive and regular graphs. In the biased case, the maximum-degree and vertex-capturing games are resolved, and we prove the clique game with $(p,q)=(1,3)$.

On edge-colouring-games by Erdős, and Bensmail and Mc Inerney

TL;DR

This work advances the study of edge-colouring games on graphs by formulating a unified framework for three Erdős- and BM-inspired games, and analyzing both unbiased and biased variants. It combines strategy-stealing arguments, combinatorial game analysis, and computer-assisted enumeration to obtain new results: a first reduction toward Erdős’ and BM’s conjectures, a strategy-stealing derivation linking different clique sizes, and a complete resolution of the biased clique game at with explicit strategies. It also investigates behaviour on Colex graphs and irregular graphs to highlight the role of symmetry, and provides a detailed description of two independent computer implementations (C and Sage) that leverage Nauty for canonical labelling and backtracking to compute optimal outcomes. The results contribute to understanding when the second player can force wins in biased settings, extend known conjectures to new graph families, and provide a practical computational toolkit for exploring edge-colouring games on broader graph classes.

Abstract

We study two games proposed by Erdős, and one game by Bensmail and Mc Inerney, all sharing a common setup: two players alternately colour edges of a complete graph, or in the biased version, they colour and edges respectively on their turns, aiming to maximise a graph parameter determined by their respective induced subgraphs. In the unbiased case, we give a first reduction towards confirming the conjecture of Bensmail and Mc Inerney, propose a conjecture for Erdős' game on maximum degree, and extend the clique and maximum-degree versions to edge-transitive and regular graphs. In the biased case, the maximum-degree and vertex-capturing games are resolved, and we prove the clique game with .
Paper Structure (14 sections, 11 theorems, 2 equations, 3 figures, 1 table)

This paper contains 14 sections, 11 theorems, 2 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Further results by strategy stealing
  3. The biased games for large $n$
  4. The biased clique game $\omega_{1,3}$
  5. Particular behaviour for some variants and extensions
  6. Smoothening cliques; the games on Colex graphs
  7. The Clique game in less symmetric situations
  8. Details about computer program
  9. General strategy
  10. Strategies for efficiency in stage one
  11. Strategies for efficiency in stage two
  12. Datastructures for memory efficiency
  13. Determining canonical labelling of a coloured graph
  14. Proof of program correctness

Key Result

Proposition 1

For every graph $G$, $s_{\omega}(G), s_{\Delta}(G) \in \{0,1\}$ and $s_{\mathop{\mathrm{VC}}\nolimits}(G) \in \{0,1,2\}$.

Figures (3)

  • Figure 1: The $5$ possible situations after $3$ edges have been coloured.
  • Figure 2: The $3$ possible situations after Bob colours its second edge.
  • Figure 3: The Colex graphs $\mathcal{C}(9)$ and $\mathcal{C}(12)$.

Theorems & Definitions (26)

  • Proposition 1
  • Conjecture 2
  • Conjecture 4: BM22
  • Proposition 5
  • Conjecture 6
  • Conjecture 7
  • Theorem 8
  • proof : Proof of \ref{['prop:n1mod4']}.
  • Proposition 9
  • proof
  • ...and 16 more