A Generalization of Some Results on List Coloring and DP-Coloring

Nakprasit, Keaitsuda Maneeruk; Nakprasit, Kittikorn

doi:10.1007/s00373-020-02177-6

Cited by 7 publications

(3 citation statements)

References 23 publications

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“…Theorem 2.2 (Nakprasit and Nakprasit [26]). Let k be an integer with k ≥ 3, and let K be an induced subgraph of G and the vertices of K can be ordered as v 1 , v 2 , .…”

Section: Some Structural Results On Critical Graphsmentioning

confidence: 99%

Variable degeneracy of graphs with restricted structures

Wang¹,

Wang²,

Yang³

2021

Preprint

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It is proved that strictly f -degenerate transversal is a generalization of list coloring, L-forested-coloring and DP-coloring. In this paper, we first give a result on strictly f -degenerate transversal of planar graphs without any configuration in Fig. 2. As a consequence, every planar graph without pairwise adjacent 3-, 4-, 5-and 6-cycles is DP-4-colorable, and has list vertex arboricity at most two. This result strengthens almost all the known results for planar graphs with DP-chromatic number four. The second main result is one on strictly f -degenerate transversal of toroidal graphs without any configuration in Fig. 3. As a consequence, every toroidal graph without pairwise adjacent 3-, 4-and 5-cycles is DP-4-colorable, and has list vertex arboricity at most two. In the final section, we present a result on strictly f -degenerate transversal of planar graphs without intersecting 5-cycles. As a consequence, every planar graph without intersecting 5-cycles is DP-4-colorable, and has list vertex arboricity at most two.

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“…Theorem 2.2 (Nakprasit and Nakprasit [26]). Let k be an integer with k ≥ 3, and let K be an induced subgraph of G and the vertices of K can be ordered as v 1 , v 2 , .…”

Section: Some Structural Results On Critical Graphsmentioning

confidence: 99%

Variable degeneracy of graphs with restricted structures

Wang¹,

Wang²,

Yang³

2021

Preprint

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“…We first extend Theorem 2.3 to the following result. Note that Theorem 3.1 was first proved in [4,Theorem 1.6], but the following proof is a little bit different from that one. Theorem 3.1.…”

Section: Strictly F -Degnerate Transversalmentioning

confidence: 96%

An extension of Thomassen's result on choosability

Li,

Wang

2021

Preprint

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Thomassen proved that every planar graph is 5-choosable. Škrekovski [8] strengthened it by showing that every K5-minor-free graph is 5-choosable. Dvořák and Postle observed that every planar graph is DP-5-colorable. In this note, we first improve these results by showing that every K5-minor-free graph is DP-5-colorable, and every K3,3-minor-free graph is DP-5-colorable. In the final section, we further improve these results under the term strictly f -degenerate transversal.

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“…DP-coloring with variable degeneracy was introduced by Nakprasit and Nakprasit [7] and Sittitrai and Nakprasit [8] as follows. Let F = (f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Relaxed DP-coloring and another generalization of DP-coloring on planar graphs without 4-cycles and 7-cycles

Nakprasit¹,

Nakprasit²,

Sittitrai³

et al. 2021

Discuss. Math. Graph Theory

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DP-coloring is generalized via relaxed coloring and variable degeneracy in [P. Sittitrai and K. Nakprasit, Sufficient conditions on planar graphs to have a relaxed DP-3-coloring, Graphs Combin. 35 (2019) 837-845], [K.M. Nakprasit and K. Nakprasit, A generalization of some results on list coloring and DP-coloring, Graphs Combin. 36 (2020) 1189-1201] and [P. Sittitrai and K. Nakprasit, An analogue of DP-coloring for variable degeneracy and its applications, Discuss. Math. Graph Theory]. In this work, we introduce another concept that includes two previous generalizations. We demonstrate its application on planar graphs without 4-cycles and 7-cycles. One implication is that the vertex set of every planar graph without 4-cycles and 7-cycles can be partitioned into three sets in which each of them induces a linear forest and one of them is an independent set. Additionally, we show that every planar graph without 4-cycles and 7-cycles is DP-(1, 1, 1)-colorable. This generalizes a result of Lih et al. [A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001) 269-273] that every planar graph without 4-cycles and 7-cycles is (3, 1) * -choosable.

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A Generalization of Some Results on List Coloring and DP-Coloring

Cited by 7 publications

References 23 publications

Variable degeneracy of graphs with restricted structures

Variable degeneracy of graphs with restricted structures

An extension of Thomassen's result on choosability

Relaxed DP-coloring and another generalization of DP-coloring on planar graphs without 4-cycles and 7-cycles

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