Every planar graph is $1$-defective $(9,2)$-paintable

Abstract

Assume $L$ is a $k$-list assignment of a graph $G$. A $d$-defective $m$-fold $L$-colouring $\varphi$ of $G$ assigns to each vertex $v$ a set $\varphi (v)$ of $m$ colours, so that $\varphi (v)\subseteq L(v)$ for each vertex $v$, and for each colour $i$, the set $\{v:i\in \varphi (v)\}$ induces a subgraph of maximum degree at most $d$. In this paper, we consider on-line list $d$-defective $m$-fold colouring of graphs, where the list assignment $L$ is given on-line, and the colouring is constructed on-line. To be precise, the $d$-defective $(k,m)$-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 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 $d$. A vertex $v$ is fully coloured if $v$ has received $m$ colours. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully coloured. Otherwise, at some round, all vertices are fully coloured and Painter wins. We say $G$ is $d$-defective $(k,m)$-paintable if Painter has a winning strategy in this game. This paper proves that every planar graph is $1$-defective $(9,2)$-paintable.

13 pages, 5 figures