Multiple list colouring of planar graphs

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.

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“…If Conjecture 1.1 is true, then the constant m needs to be at least 2. The following conjecture (also posed in [15]) says that 2 is enough.…”

Section: Introductionmentioning

confidence: 99%

Locally planar graphs are 2-defective 4-paintable

Han

Zhu

2016

European Journal of Combinatorics

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“…The integer m depends on G and is usually a large integer. A natural question is for which (a, b), G is (am, bm)-choosable for any positive integer m. This motivated the definition of strong fractional choice number of a graph [5].…”

Section: Introductionmentioning

confidence: 99%

“…In any case, ch * f (G) is an interesting graph invariant and there are many challenging problems concerning this parameter. The strong fractional choice number of planar graphs were studied in [5] and [4]. Let P denote the family of planar graphs and for a positive integer k, let P k be the family of planar graphs containing no cycles of length k. It was proved in [5] that 5 ≥ ch * f (P) ≥ 4 + 2 9 and prove in [4] that 4 ≥ ch * f (P 3 ) ≥ 3 + 1 17.…”

Section: Introductionmentioning

confidence: 99%

The strong fractional choice number of series–parallel graphs

Zhu

2020

Discrete Mathematics

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The strong fractional choice number of a graph G is the infimum of those real numbers r such that G is (⌈rm⌉, m)-choosable for every positive integer m. The strong fractional choice number of a family G of graphs is the supremum of the strong fractional choice number of graphs in G. We denote by Q k the class of series-parallel graphs with girth at least k. This paper proves that for k = 4q − 1, 4q, 4q + 1, 4q + 2, the strong fractional number of Q k is exactly 2 + 1 q .

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“…So is the problem as to determine (or to find bounds for) the infimum of those

italicϵ

for which every triangle‐free planar graph is

(3 + italicϵ)

‐choosable. The concept of strong fractional choice number studied in and the concept of

λ

‐choosability studied in provide refinements of choice number from different prospectives.…”

Section: Introductionmentioning

confidence: 99%

“…For this question to be meaningful, we need some refinements of the choice number of a graph. A few such refinements and variations are studied recently [5,6,[12][13][14].Definition 1. Assume G is a graph, k is a positive integer and X is a subset of V G ( ).…”

mentioning

confidence: 99%