Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

Planar graphs without 4‐cycles are acyclically 6‐choosable

A proper vertex coloring of a graph G=(V, E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L={L(v)|v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L‐list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k‐choosable. In this paper we prove that every planar graph G without 4‐cycles is acyclically 6‐choosable. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 307–323, 2009     

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 with neither 5‐cycles nor close 3‐cycles are 3‐colorable

In 2003, O. V. Borodin and A. Raspaud conjectured that every planar graph with the minimum distance between triangles, d_Δ, at least 1 and without 5‐cycles is 3‐colorable. This Bordeaux conjecture (Brdx3CC) has common features with Havel's (1970) and Steinberg's (1976) 3‐color problems. A relaxation of Brdx3CC with d_Δ?4 was confirmed in 2003 by Borodin and Raspaud, whose result was improved to d_Δ?3 by Borodin and A. N. Glebov (2004) and, independently, by B. Xu (2007). The purpose of this article is to make the penultimate step (d_Δ?2) towards Brdx3CC. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 1–31, 2010     

Acyclic 5-choosability of planar graphs without 4-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

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).     

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

不含5-圈的平面图的无圈边着色

图G的一个无圈边着色是一个正常的边着色且不含双色的圈.图G的无圈边色数是图G的无圈边着色中所用色数的最小者.本文用反证法得到了不含5-圈的平面图G的无圈边色数的一个上界.     

不含5-圈和相邻6-圈的平面图的全染色

图的正常k-全染色是用k种颜色给图的顶点和边同时进行染色,使得相邻或者相关联的元素(顶点或边)染不同的染色.使得图G存在正常k-全染色的最小正整数k,称为图G的全色数,用χ″(G)表示.证明了若图G是最大度△≥6且不含5-圈和相邻6-圈的平面图,则χ″(G)=△+1.     

On 3-colorability of planar graphs without adjacent short cycles

A short cycle means a cycle of length at most 7.In this paper,we prove that planar graphs without adjacent short cycles are 3-colorable.This improves a result of Borodin et al.(2005).     

Planar graphs without adjacent cycles of length at most seven are 3-colorable

We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3≤i≤j≤7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs.     

Star coloring bipartite planar graphs

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009     

Group edge choosability of planar graphs without adjacent short cycles

In this paper,we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group(Δ(G)+1)-edge-choosable,and some planar graphs with large girth and maximum degree are groupΔ(G)-edge-choosable.     

Smaller planar triangle-free graphs that are not 3-list-colorable

In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors.