Locally planar graphs are 2-defective 4-paintable

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Abstract

The d-defective k-painting game on a graph G is played by two players: Lister and Painter. Initially, each vertex has k tokens and is uncoloured. In each round, Lister chooses a set M of uncoloured vertices and removes one token from each chosen vertex. Painter colours a subset X of M which induces a subgraph G[X] of maximum degree at most d. Lister wins the game if at the end of some round, a vertex v has no more tokens left, and is uncoloured. Otherwise, at some round, all vertices are coloured and Painter wins. We say G is d-defective k-paintable if Painter has a winning strategy in this game. This paper proves that for each surface S, there is a constant w such that graphs embedded in S with edge-width at least w are 2-defective 4-paintable.

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