DP-3-Coloring of Planar Graphs Without 4, 9-Cycles and Cycles of Two Lengths from 
                
                  
                
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Liu, Runrun; Loeb, Sarah; Rolek, Martin; Yin, Yuxue; Yu, Gexin

doi:10.1007/s00373-019-02025-2

Cited by 18 publications

(5 citation statements)

References 12 publications

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“…This strengthens the results that every planar graph without 4-, 5-, 6-, and 9-cycles is 3-DPcolorable [8], and every planar graph without 4-, 6-, 7-, and 9-cycles is 3-DP-colorable [7].…”

Section: An Application Of the Operationsupporting

confidence: 86%

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Weak degeneracy of planar graphs without 4- and 6-cycles

Wang

2023

Preprint

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A graph is k-degenerate if every subgraph H has a vertex v with d H (v) ≤ k. The class of degenerate graphs plays an important role in the graph coloring theory. Observed that every k-degenerate graph is (k + 1)-choosable and (k + 1)-DP-colorable. Bernshteyn and Lee defined a generalization of k-degenerate graphs, which is called weakly k-degenerate. The weak degeneracy plus one is an upper bound for many graph coloring parameters, such as choice number, DP-chromatic number and DP-paint number. In this paper, we give two sufficient conditions for a plane graph without 4-and 6-cycles to be weakly 2-degenerate, which implies that every such graph is 3-DP-colorable and near-bipartite, where a graph is near-bipartite if its vertex set can be partitioned into an independent set and an acyclic set.

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“…This strengthens the results that every planar graph without 4-, 5-, 6-, and 9-cycles is 3-DPcolorable [8], and every planar graph without 4-, 6-, 7-, and 9-cycles is 3-DP-colorable [7].…”

Section: An Application Of the Operationsupporting

confidence: 86%

“…If v is incident with exactly two 5-faces, then the other incident face is an 8 + -face by Lemma 2.2(8), and then µ ′ (v) = −2 × 1 8 + 1 4 = 0 R2c. Since there are no 9-cycles, v cannot be incident with three 5-faces by Lemma 2.2(7). If v is incident with three 6 + -faces, then µ ′ (f ) = µ(f ) ≥ 0.Let v be a 4-vertex.…”

mentioning

confidence: 99%

Weak degeneracy of planar graphs without 4- and 6-cycles

Wang

2023

Preprint

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“…It is difficult to improve Theorem for small values of

k

. Recently, Liu, the present author, and Yu have been able to show that for

(k_{0}, m) \in {(15, 8), (27, 9), (43, 10)}

, and

k \geq k_{0}

, any

k

‐contraction‐critical graph is

m

‐connected. Some improvements for larger

k

have also been found, for example .…”

Section: Introductionmentioning

confidence: 92%

Graphs with no K9= minor are 10‐colorable

Rolek

2019

Journal of Graph Theory

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Hadwiger's conjecture claims that any graph with no K t minor is ( t − 1 )‐colorable. This has been proved for t ≤ 6, but remains open for t ≥ 7. As a variant of this conjecture, graphs with no K t = minor have been considered, where K t = denotes the complete graph with two edges removed. It has been shown that graphs with no K t = minor are ( 2 t − 8 )‐colorable for t ∈ { 7 , 8 }. In this paper, we extend this result to the case t = 9 and show that graphs with no K 9 = minor are 10‐colorable.

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“…This proves Formula (9). We will verify the negativeness of the final charge of H by using Formulas (8) and (9). Firstly, assume that H contains a 2-vertex, say v j .…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Denote by △ d the smallest distance between triangles. The DP-3-colorability was proved for planar graphs with ≥ △ d 3 and with no cycle of length from {4, 5} [17], planar graphs with ≥ △ d 2 and with no cycle of length from {4, 5, 6} [17], {4, 5, 7} [13], {5, 6, 7} [10], or {5, 6, 8} [13], planar graphs with neither intersecting triangles nor cycle of length from {4, 5, 6, 7} [12], planar graphs with neither adjacent triangles nor cycle of length from {5, 6, 9} [13], and planar graphs with no cycle of length from {3, 5, 6} [10], {3, 6, 7, 8} [10], {4, 5, 6, 9} [10], {4, 5, 7, 9} [10], {4, 6, 7, 9} [9], {4, 6, 8, 9} [9], or {4, 7, 8, 9} [9].…”

Section: Introductionmentioning

confidence: 99%