DP-4-colorability of planar graphs without intersecting 5-cycles

DP-coloring of graphs as a generalization of list coloring was introduced by Dvo?ák and Postle (2018). In this paper, we show that every planar graph without intersecting 5-cycles is DP-4-colorable, which improves the result of Hu and Wu (2017), who proved that every planar graph without intersecting 5-cycles is 4-choosable, and the results of Kim and Ozeki (2018).     

DP-4-colorability of two classes of planar graphs

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvo?ák and Postle in 2017. In this paper, we prove that every planar graph 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 vertices in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.     

Note on list star edge-coloring of subcubic graphs

A star edge-coloring of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most 7, answering the question of Dvořák et al (J Graph Theory, 72 (2013), 313-326).     

A note on a Brooks' type theorem for DP-coloring

Dvořák and Postle introduced DP-coloring of simple graphs as a generalization of list-coloring. They proved a Brooks' type theorem for DP-coloring; and Bernshteyn, Kostochka, and Pron extended it to DP-coloring of multigraphs. However, detailed structure, when a multigraph does not admit DP-coloring, was not specified. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.     

Sharp Dirac's theorem for DP‐critical graphs

Correspondence coloring, or DP‐coloring, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that satisfy Dirac's bound with equality.     

无8-,9-和10-圈的平面图的3-可选择性

寻找平面图是3-或者4-可选择的充分条件是图的染色理论中一个重要研究课题,本文研究了围长至少是4的特殊平面图的选择数,通过权转移的方法证明了每个围长至少是4且不合8-圈,9-圈和10-圈的平面图是3-可选择的．     

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem [4] states that every planar triangle-free graph is 3-colorable. However, Voigt [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.     

Planar graphs without cycles of length 4, 5, 8, or 9 are 3-choosable

It is shown that a planar graph without cycles of length 4, 5, 8, or 9 is 3-choosable.     

Acyclic 3-choosability of planar graphs without cycles of length from 4 to 12

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (O. V. Borodin et al., 2002). This conjecture if proved would imply both Borodin’s acyclic 5-color theorem (1979) and Thomassen’s 5-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-colorable. In particular, a planar graph of girth at least 7 is acyclically 3-colorable (O. V. Borodin, A. V. Kostochka and D. R. Woodall, 1999) and acyclically 3-choosable (O. V. Borodin et. al, 2009). A natural measure of sparseness, introduced by Erdős and Steinberg, is the absence of k-cycles, where 4 ≤ k ≤ S. Here, we prove that every planar graph without cycles of length from 4 to 12 is acyclically 3-choosable.     

Acyclic 4-choosability of planar graphs with neither 4-cycles nor triangular 6-cycles

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (Borodin et al. 2002) [7]. This conjecture if proved would imply both Borodin’s acyclic 5-color theorem (1979) and Thomassen’s 5-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs.Some sufficient conditions are also obtained for a planar graph to be acyclically 4-choosable and 3-choosable. In particular, acyclic 4-choosability was proved for the following planar graphs: without 3-cycles and 4-cycles (Montassier, 2006 [23]), without 4-cycles, 5-cycles and 6-cycles (Montassier et al. 2006 [24]), and either without 4-cycles, 6-cycles and 7-cycles, or without 4-cycles, 6-cycles and 8-cycles (Chen et al. 2009 [14]).In this paper it is proved that each planar graph with neither 4-cycles nor 6-cycles adjacent to a triangle is acyclically 4-choosable, which covers these four results.     

Note on 3-choosability of planar graphs with maximum degree 4

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.     

Every planar graph without 4-cycles adjacent to two triangles is DP-4-colorable

Wang and Lih (2002) conjectured that every planar graph without adjacent triangles is 4-choosable. In this paper, we prove that every planar graph without any 4-cycle adjacent to two triangles is DP-4-colorable, which improves the results of Lam et al. (1999), Cheng et al. (2016) and Kim and Yu [ arXiv:1709.09809v1].     

A note on the acyclic 3-choosability of some planar graphs

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v∈V(G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring ? of G such that ?(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment L with |L(v)|≥k for all v∈V(G), then G is said to be acyclically k-choosable. Borodin et al. proved that every planar graph with girth at least 7 is acyclically 3-choosable (Borodin et al., submitted for publication [4]). More recently, Borodin and Ivanova showed that every planar graph without cycles of length 4 to 11 is acyclically 3-choosable (Borodin and Ivanova, submitted for publication [7]). In this note, we connect these two results by a sequence of intermediate sufficient conditions that involve the minimum distance between 3-cycles: we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less than 7 (resp. 5, 3, 2) is acyclically 3-choosable.     

Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

It is known that planar graphs without cycles of length 4, i, j, or 9 with 4<i<j<9, except that i=7 and j=8, are 3-choosable. This paper proves that planar graphs without cycles of length 4, 7, 8, or 9 are also 3-choosable.     

没有4至6-圈的平面图是(1,0,0)-可染的

设d₁, d₂,..., d_k 是k个非负整数. 若图G=(V,E) 的顶点集V可剖分成k个子集V₁, V₂,..., V_k,使得对i=1, 2,..., k 由V_i 所导出的子图G[V_i] 的最大度至多为d_i, 则称G是(d₁, d₂,..., d_k)-可染的. 著名的Steinberg 猜想断言, 每个既没有4-圈又没有5-圈的平面图是(0, 0, 0)-可染的. 对此猜想已经证明每个没有4 至7-圈的平面图是(0, 0, 0)-可染的, 但还没有发现有人证明每个没有4 至6-圈的平面图是(0, 0, 0)-可染的. 本文证明没有4 至6-圈的平面图是(1, 0, 0)-可染的.     

Planar graphs without 4‐cycles adjacent to 3‐cycles are list vertex 2‐arborable

It is known that not all planar graphs are 4‐choosable; neither all of them are vertex 2‐arborable. However, planar graphs without 4‐cycles and even those without 4‐cycles adjacent to 3‐cycles are known to be 4‐choosable. We extend this last result in terms of covering the vertices of a graph by induced subgraphs of variable degeneracy. In particular, we prove that every planar graph without 4‐cycles adjacent to 3‐cycles can be covered by two induced forests. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 234–240, 2009     

The Johansson‐Molloy theorem for DP‐coloring

The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound for triangle‐free graphs G, avoiding the technicalities of the entropy compression method and only using the usual “lopsided” Lovász Local Lemma (albeit in a somewhat unusual setting). On the other hand, we extend Molloy's result to DP‐coloring (also known as correspondence coloring), a generalization of list coloring introduced recently by Dvo?ák and Postle.