Proper colouring Painter–Builder game

Małgorzata Bednarska-Bzdęga, Michael Krivelevich, Viola Mészáros, Clément Requilé

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)658-664
Number of pages7
JournalDiscrete Mathematics
Issue number3
StatePublished - Mar 2018


  • Combinatorial games
  • Graph colouring


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