Locally planar graphs are 2-defective 4-paintable

Abstract

Assume L is a k-list assignment of a graph G. A d-defective m-fold L-colouring φ of G assigns to each vertex v a set φ(v) of m colours, so that φ(v) ⊆ L(v) for each vertex v, and for each colour i, the set {v : i ∈ φ(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.

Defective 3-Paintability of Planar Graphs

“…As mentioned above, it is known that every planar graph is 2-defective 3choosable [3,8] and 1-defective 4-choosable [2]. Recently, Han and Zhu [4] proved that every planar graph is 2-defective 4-paintable. It remained open questions whether or not every planar graph is 2-defective 3-paintable, or 1defective 4-paintable.…”

The Alon-Tarsi number of a planar graph minus a matching

“…On the other hand, it is known that every planar graph is 5-paintable [12], 3-defective 3-paintable [7], and 2-defective 4-paintable [8]. For defective paintability of the family of planar graphs, Question 2 below is the only question remained open.…”

Multiple list colouring of planar graphs

“…It is proved recently in [4] that every planar graph is 1-defective (9, 2)-paintable, which implies that every planar graph is 1-defective (9, 2)-choosable (i.e., if each vertex has 9 permissible colours, then there is a 2-fold colouring of the vertices of G with permissible colours so that each colour class induces a graph of maximum degree at most 1.) The following conjecture, which is stronger than Conjecture 3, asserts that the 1-defective can be replaced by 0-defective.…”