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arXiv · 2505.03497

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

Stijn Cambie·Michiel Provoost
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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 $(p,q)=(1,3)$.

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Stijn Cambie, Michiel Provoost. 2025-10-28. On edge-colouring-games by Erdős, and Bensmail and Mc Inerney. https://arxiv.org/abs/2505.03497

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