A sufficient condition for DP-4-colorability

Kim, Seog‐Jin; Ozeki, Kenta

doi:10.1016/j.disc.2018.03.027

Cited by 45 publications

(40 citation statements)

References 23 publications

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“…More specifically, we are interested in finding sufficient conditions for a planar graph to be DP -4-colorable. The next result of Kim and Ozeki [5] provides an important motivation for our research. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 92%

DP-4-colorability of two classes of planar graphs

Chen

Liu

et al. 2019

Discrete Mathematics

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DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). In this paper, we prove that every planar graph G without 4-cycles adjacent to k-cycles is DP-4-colorable for k = 5 and 6. As a consequence, we obtain two new classes of 4-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.Recently, Dvorák and Postle [2] introduced DP -coloring (also known as correspondence coloring) as a generalization of list coloring.Definition 1.1. Let G be a simple graph with n vertices and let L be a list assignment for V (G). For each edge uv in G, let M uv be a matching between the sets L(u) and L(v) and let M L = {M uv : uv ∈ E(G)}, called the matching assignment. Let H L be the graph that satisfies the following conditions

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Section: Introductionmentioning

confidence: 92%

DP-4-colorability of two classes of planar graphs

Chen

Liu

et al. 2019

Discrete Mathematics

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“…Kim and Ozeki [6] showed that planar graphs without 4-cycles are DP-4-colorable. Some more sufficient conditions for a planar graph to be DP -4-colorable have been found in [2,6,7,9,10], and we summarize them below. 6,7,9,10]) The following planar graphs are DP-4-colorable (3,5), (3,6), (4,5), (4, 6)}.…”

Section: Introductionmentioning

confidence: 99%

“…Case 4.1. H 4 is isomorphic to one of Figure 1 (4)- (6). By (R1), H 4 gets 1 2 · 4, 1 2 · 6, 1 2 · 6 from adjacent 8 + -faces in (4), (5), (6), respectively.…”

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confidence: 99%

“…H 4 is isomorphic to one of Figure 1 (4)- (6). By (R1), H 4 gets 1 2 · 4, 1 2 · 6, 1 2 · 6 from adjacent 8 + -faces in (4), (5), (6), respectively. If any vertex in H 4 is on C, then by (R5), H 4 gets at least 2, 1, 1 from C in (4), (5), (6) respectively, so µ * (H 4 ) ≥ 0.…”

mentioning

confidence: 99%

“…By (R1), H 4 gets 1 2 · 4, 1 2 · 6, 1 2 · 6 from adjacent 8 + -faces in (4), (5), (6), respectively. If any vertex in H 4 is on C, then by (R5), H 4 gets at least 2, 1, 1 from C in (4), (5), (6) respectively, so µ * (H 4 ) ≥ 0. If H 4 is internal, then by (R2), in (4) it gets 1 2 from each of u, v, w, x, in (5) it gets 1 2 from each of v and y, and in (6), it gets 1 2 from each of v, w, x.…”

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confidence: 99%

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Planar graphs without 7-cycles and butterflies are DP-4-colorable

Kim

Liu

2020

Discrete Mathematics

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DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořák and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-4-colorable. In particular, for each k ∈ {3, 4, 5, 6}, every planar graph without k-cycles is DP-4-colorable. In this paper, we prove that every planar graph without 7-cycles and butterflies is DP-4-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters formed by many triangles.

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