Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

Yin, Yuxue; Yu, Gexin

doi:10.48550/arxiv.1809.00925

Cited by 1 publication

(2 citation statements)

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“…Liu and Li [5], Sittitrai and Nakprasit [10], Kim and Yu [8], and Kim and Ozeki [7] all extend results on 4-choosability of planar graphs to DP-4-coloring. Yin and Yu [16] extend results for 3-choosability to DP-3-coloring, in some cases for a larger class of graphs than the analogous choosability result. Among other results extending that conditions for 3-choosability hold for DP-3-coloring, Liu, Loeb, Yin, and Yu [6] show that planar graphs with no {4, 5, 6, 9}-cycles or with no {4, 5, 7, 9}-cycles are DP-3-colorable.…”

Section: Introductionmentioning

confidence: 57%

“…Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring [5,6,7,8,10,16]. We note that list-coloring results do not always extend to DP-coloring results, as shown in [2].…”

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confidence: 92%

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DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$

Liu,

Loeb,

Rolek

et al. 2018

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A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle [3]. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring [5,6,7,8,10,16]. We note that list-coloring results do not always extend to DP-coloring results, as shown in [2]. Our main result in this paper is to prove that every planar graph without cycles of length {4, a, b, 9} for a, b ∈ {6, 7, 8} is DP-3-colorable, extending three existing results [9, 13, 14] on 3-choosability of planar graphs.Definition 1.1. Let G be a simple graph with n vertices and let L be a list assignment for G. For each v ∈ V (G), let L v = {v} × L(v). For each edge uv ∈ E(G), let M uv be a matching

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