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 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 kcoloured. 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. ∗Corresponding author. Email addresses: [email protected] (Małgorzata Bednarska-Bzdęga), [email protected] (Michael Krivelevich), [email protected] (Viola Mészáros), [email protected] (Clément Requilé) 1Supported in part by USA-Israel BSF grant 2014361 and by ISF grant 1261/17. 2Supported by Swiss National Science Foundation grants 200020-162884 and 200020144531. 3Supported by the Austrian Science Fund (FWF) grant F5004. The work presented in this paper was carried out while the author was affiliated with the Institut für Mathematik und Informatik, Freie Universität Berlin, and Berlin Mathematical School, Germany, and supported by the FP7-PEOPLE-2013-CIG project CountGraph (ref. 630749). Preprint submitted to Elsevier October 16, 2017