DP-4-colorability of two classes of planar graphs

Abstract

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

Variable degeneracy of graphs with restricted structures

“…This strengthens all the results in [2,11,16,21,23,28] for DP-coloring and the results in [7,9,10,30] for (list) vertex arboricity.…”

“…Then there exists a vertex w adjacent to each of w 1 , w 2 , w 3 and w 4 . By (2), each of [ww 1 w 2 ], [ww 2 w 3 ], [ww 3 w 4 ] and [ww 4 w 1 ] bounds a 3-face. We will say a region bounded by [w 1 w 2 w 3 w 4 ] consists of four 3-faces.…”

“…It is proved [25] that strictly f -degenerate transversal is a common generalization of list coloring, Lforested-coloring and DP-coloring. It is proved that the following classes of graphs have DP-chromatic number at most four: planar graphs without 3-cycles adjacent to 4-cycles [16]; planar graphs without 3-cycles adjacent to 5-cycles [23]; planar graphs without 3-cycles adjacent to 6-cycles [23]; planar graphs without 4cycles adjacent to 5-cycles [2]; planar graphs without 4-cycles adjacent to 6-cycles [2]; planar graphs without 5-cycles adjacent to 6-cycles [21]; planar graphs without pairwise adjacent 3-, 4-and 5-cycles [28].…”

Planar Graphs with the Distance of 6--Cycles at Least 2 from Each Other Are DP-3-Colorable

“…Bernshteyn et al [12][13][14][15][16] gave some results on DP-coloring. DP-3-colorable planar graphs can be found in [17,18] and DP-4-colorable planar graphs can be found in [19][20][21]. Yin and Yu [22] proved planar graphs with no {4, 5, 6}-cycles in which the distance between triangles is at least 2 are DP-3-colorable.…”

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

“…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)}.…”