Proper colouring Painter-Builder game

TL;DR

This paper analyzes a two-player graph colouring game where Painter and Builder alternate moves, establishing that the minimum number of colours needed for Painter to guarantee a win grows logarithmically with the number of vertices.

Contribution

The paper introduces and analyzes the proper colouring Painter-Builder game, proving that the minimal number of colours for Painter's win scales logarithmically with graph size.

Findings

Minimal colours for Painter's win are logarithmic in n.

Biased versions of the game are also studied.

Painter can guarantee a win with O(log n) colours.

Abstract

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 claims an edge between two previously unconnected vertices. Both players should maintain that during the game the graph admits a proper $k$-colouring. 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.