On edge-colouring-games by Erd\H{o}s, and Bensmail and Mc Inerney

TL;DR

This paper investigates edge-colouring games on graphs, providing new reductions, conjectures, and solutions for various game variants, including biased and unbiased versions, on different classes of graphs.

Contribution

It offers the first reduction towards Erdős and Bensmail-McInerney conjectures, proposes a new conjecture, and solves several game variants including biased cases.

Findings

Reduced the problem towards Erdős and Bensmail-Mc Inerney conjectures.

Proposed a new conjecture for Erdős' game on maximum degree.

Solved the clique game with parameters (p,q)=(1,3).

Abstract

We study two games proposed by Erd\H{o}s, 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 Erd\H{o}s' 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 $(p,q)=(1,3)$.