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

Ming Han, H. A. Kierstead, Xuding Zhu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Assume G is a graph, and k,d,m are natural numbers. The d-defective (k,m)-painting game on G is played by two players: Lister and Painter. Initially, each vertex has k tokens and is uncolored. In each round, Lister chooses a set M of vertices and removes one token from each chosen vertex. Painter colors a subset X of M which induces a subgraph G[X] of maximum degree at most d. A vertex v is fully colored if v has received m colors. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully colored. Otherwise, at some round, all vertices are fully colored and Painter wins. We say G is d-defective (k,m)-paintable if Painter has a winning strategy in this game. We prove that every planar graph is 1-defective (9,2)-paintable. In addition, our winning strategy also implies that every planar graph is 1-defective [Formula presented]-paintable for any positive m.

Original languageEnglish (US)
Pages (from-to)257-264
Number of pages8
JournalDiscrete Applied Mathematics
StatePublished - May 15 2021


  • On-line list coloring
  • Planar graph
  • d-defective painting game

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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