Defective 3-Paintability of 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 ddefective 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.

The Alon-Tarsi number of subgraphs of a planar graph

“…Then, combining S j for j ∈ [6], we obtain a subgraph of S−E(H) isomorphic to a member of J. This completes the proof.…”

“…Let J be the set of graphs obtained from the disjoint union of 6 copies of J 1 or J 2 by identifying the edges corresponding to ab from each copy. For each G ∈ J, let c i , d i , e i be the vertices corresponding to c, d, e, respectively, for i ∈ [6], and the edge ab in G is called the handle of G. (See Figure 2.)…”

“…. < i 6 ≤ 9 such that every edge incident with b in S i j is not contained in H for j ∈ [6]. Without loss of generality, let i j = j for j ∈ [6].…”

“…< i 6 ≤ 9 such that every edge incident with b in S i j is not contained in H for j ∈ [6]. Without loss of generality, let i j = j for j ∈ [6]. Then, by Lemma 2.2, for each j ∈ [6], S j − E(H) contains K 4 or a subgraph isomorphic to J 1 or J 2 with handle ab.…”

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

“…But Painter has a winning strategy, so, he will eventually obtain a ddefective L-colouring of G. The converse is however not true. Though all planar graphs are 2-defective 3-choosable, an example of non-2-defective 3-paintable planar graph has been constructed in [7].…”