Analogue of DP-coloring on variable degeneracy and its applications on list vertex-arboricity and DP-coloring

Sittitrai, Pongpat; Nakprasit, Kittikorn

doi:10.48550/arxiv.1807.00815

Cited by 2 publications

(2 citation statements)

References 21 publications

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“…Theorem 5.10 (Sittitrai and Nakprasit [49]). Let G be a planar graph without 4-cycles adjacent to 3-cycles.…”

Section: Corollarymentioning

confidence: 99%

Cover and variable degeneracy

Lu,

Wang,

Wang

2019

Preprint

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Let f be a nonnegative integer valued function on the vertex-set of a graph. A graph is strictly f -degenerate if each nonempty subgraph Γ has a vertex v such that deg Γ (v) < f (v). In this paper, we define a new concept, strictly f -degenerate transversal, which generalizes list coloring, ( f 1 , f 2 , . . . , f κ )-partition, signed coloring, DP-coloring andThe main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable result, signed degree-colorable result, and DP-degree-colorable result. Similar to Borodin, Kostochka and Toft's variable degeneracy, this degree type result is also self-strengthening. We also give some structural results on critical graphs with respect to strictly f -degenerate transversal. Using these results, we can uniformly prove many new and known results. In the final section, we give some open problems.

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“…Theorem 5.10 (Sittitrai and Nakprasit [49]). Let G be a planar graph without 4-cycles adjacent to 3-cycles.…”

Section: Corollarymentioning

confidence: 99%

Cover and variable degeneracy

Lu,

Wang,

Wang

2019

Preprint

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show abstract

“…The concept of variable degeneracy seems to have firstly been used by Borodin, Kostochka, and Toft [6]. DP-colorings with variable degeneracy where introduced by Sittitrai and Nakprasit [33] although they use a slightly different approach.…”

Section: Dp-coloring and Variable Degeneracymentioning

confidence: 99%

Generalized DP-Colorings of Graphs

Kostochka¹,

Schweser²,

Stiebitz³

2019

Preprint

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Given a graph property P, a P-coloring of a graph G with color set C is a mapping ϕ : V (G) → C such that for each color c ∈ C the subgraph of G induced by the color class ϕ −1 (c) belongs to P. The P-chromatic number χ(G : P) of G is the least number k for which G admits a P-coloring with a set of k-colors. This coloring concept dates back to the late 1960s and is commonly known as generalized coloring. In the 1980s the P-choice number χ ℓ (G : P) of G was introduced and investigated by several authors. In 2018 Dvořák and Postle introduced the DP-chromatic number as a natural extension of the choice number. They also remarked that this concept applies to any graph property. This motivated us to investigate the P-DP-chromatic number χ DP (G : P) of G. We have χ(G : P) ≤ χ ℓ (G : P) ≤ χ DP (G : P). In this paper we show that various fundamental coloring results, in particular, the theorems of Brooks, of Gallai, and of Erdős, Rubin and Taylor, have counterparts for the P-DP-chromatic number.

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Analogue of DP-coloring on variable degeneracy and its applications on list vertex-arboricity and DP-coloring

Cited by 2 publications

References 21 publications

Cover and variable degeneracy

Cover and variable degeneracy

Generalized DP-Colorings of Graphs

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