DP-3-coloring of some planar graphs

Liu, Runrun; Loeb, Sarah; Yin, Yuxue; Yu, Gexin

doi:10.48550/arxiv.1802.09312

Cited by 5 publications

(11 citation statements)

References 13 publications

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“…One may think of A(v) as the colors available at v after coloring G − H. Lemma 2.2. [6] Let k ≥ 3 and H be a subgraph of G. If the vertices of H can be ordered as v 1 , v 2 , . .…”

Section: Lemmas and A Brief Discussion Of The Dischargingmentioning

confidence: 99%

“…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. In this paper, we extend the three previous results for 3-choosability of planar graphs stated in Theorem 1.2.…”

Section: Introductionmentioning

confidence: 93%

“…Such a structure is not necessarily reducible in the setting of DP-coloring. In Section 2, we use the lemma about "near-(k − 1)-degenerate" subgraphs from [6] which fills a similar role in our reducible structures. In Section 2, we also provide our reducible structures and a lemma about how much charge can be given by large faces in our subsequent discharging arguments.…”

Section: Theorem 12 a Planar Graph G Is 3-choosable If One Of The Fol...mentioning

confidence: 99%

“…Section 3 provides the proofs for Theorem 1.3. We use different initial charges from the ones in [6], and provide a new unified set of discharging rules for all three cases.…”

Section: Theorem 12 a Planar Graph G Is 3-choosable If One Of The Fol...mentioning

confidence: 99%

“…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].…”

mentioning

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

Preprint

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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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“…One may think of A(v) as the colors available at v after coloring G − H. Lemma 2.2. [6] Let k ≥ 3 and H be a subgraph of G. If the vertices of H can be ordered as v 1 , v 2 , . .…”

Section: Lemmas and A Brief Discussion Of The Dischargingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 93%

Section: Theorem 12 a Planar Graph G Is 3-choosable If One Of The Fol...mentioning

confidence: 99%

“…Section 3 provides the proofs for Theorem 1.3. We use different initial charges from the ones in [6], and provide a new unified set of discharging rules for all three cases.…”

Section: Theorem 12 a Planar Graph G Is 3-choosable If One Of The Fol...mentioning

confidence: 99%

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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

Preprint

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

Yin

2019

Discrete Mathematics

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Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers d 0 and d 1 such that planar graphs without {4, 5}-cycles and d ∆ ≥ d 0 are 3-choosable and planar graphs without {4, 5, 6}-cycles and d ∆ ≥ d 1 are 3-choosable, where d ∆ is the smallest distance between triangles. They showed that 2 ≤ d 0 ≤ 4 and d 1 ≤ 3. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without {4, 5}-cycles and d ∆ ≥ 3 are DP-3-colorable, and (2) planar graphs without {4, 5, 6}-cycles and d ∆ ≥ 2 are DP-3-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, d 0 ≤ 3 and d 1 ≤ 2. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.

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Every planar graph without adjacent cycles of length at most $8$ is $3$-choosable

Liu¹,

Li²

2018

Preprint

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DP-3-coloring of some planar graphs

Cited by 5 publications

References 13 publications

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

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

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

Every planar graph without adjacent cycles of length at most $8$ is $3$-choosable

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