# Defective 3-Paintability of Planar Graphs

## TL;DR

This paper investigates a graph coloring game called defective painting, proving that all planar graphs can be colored with certain defect and token constraints, but some cannot with others, highlighting limitations in planar graph colorability.

## Contribution

It establishes that every planar graph is 3-defective 3-paintable and provides a construction of a planar graph that is not 2-defective 3-paintable.

## Key findings

- All planar graphs are 3-defective 3-paintable.
- There exists a planar graph not 2-defective 3-paintable.
- The study advances understanding of defective coloring in planar graphs.

## Abstract

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.