Proper colouring Painter–Builder game

We consider the following two-player game, parametrised by positive integers n and k. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on n vertices. In each round Painter colours a vertex of her choice by one o...

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Bibliographic Details
Published inDiscrete mathematics Vol. 341; no. 3; pp. 658 - 664
Main Authors Bednarska-Bzdęga, Małgorzata, Krivelevich, Michael, Mészáros, Viola, Requilé, Clément
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.03.2018
Subjects
Combinatorial games
Graph colouring
Combinatorial games
Graph colouring
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Summary:We consider the following two-player game, parametrised by positive integers n and k. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on n vertices. In each round Painter colours a vertex of her choice by one of the k colours and Builder adds an edge between two previously unconnected vertices. Both players must adhere to the restriction that the game graph is properly k-coloured. The game ends if either all n vertices have been coloured, or Painter has no legal move. In the former case, Painter wins the game; in the latter one, Builder is the winner. We prove that the minimal number of colours k=k(n) allowing Painter’s win is of logarithmic order in the number of vertices n. Biased versions of the game are also considered.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2017.11.008