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.     

Equitable list coloring of planar graphs without 4- and 6-cycles

A graph G is equitably k-choosable if for any k-uniform list assignment L, there exists an L-colorable of G such that each color appears on at most vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for k>Δ(G). We prove this for planar graphs with Δ(G)≥6 and no 4- or 6-cycles.     

Total colorings and list total colorings of planar graphs without intersecting 4-cycles

Suppose that G is a planar graph with maximum degree Δ and without intersecting 4-cycles, that is, no two cycles of length 4 have a common vertex. Let χ^″(G), and denote the total chromatic number, list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that χ^″(G)=Δ+1 if Δ≥7, and and if Δ(G)≥8. Furthermore, if G is a graph embedded in a surface of nonnegative characteristic, then our results also hold.     

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.     

Acyclic 5-choosability of planar graphs with neither 4-cycles nor chordal 6-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 ? of G such that ?(v)∈L(v) for all v∈V(G). 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 it is proved that every planar graph with neither 4-cycles nor chordal 6-cycles is acyclically 5-choosable. This generalizes the results of [M. Montassier, A. Raspaud, W. Wang, Acyclic 5-choosability of planar graphs without small cycles, J. Graph Theory 54 (2007) 245-260], and a corollary of [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (4) (2006) 281-300].     

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).     

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.     

Total coloring of planar graphs without chordal 7-cycles

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ(G) ≥ 7 and without chordal 7-cycles,then G has a(Δ(G) + 1)-total-coloring.     

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exists a proper edge-coloring ? of G such that ?(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ(G)≠5 and without adjacent 3-cycles, or with maximum degree Δ(G)≠5,6 and without 7-cycles, then G is edge-(Δ(G)+1)-choosable.     

A note on the Three Color Problem on planar graphs without 4- and 5-cycles and without ext-triangular 7-cycles

Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. This conjecture is disproved by constructing a planar graph with no cycles of length four or five but intersecting triangles. Jin et al. proved that plane graphs without 4- and 5-cycles and without ext-triangular 7-cycles are 3-colorable [SIAM J. Discrete Math. 31 (3) (2017) 1836–1847]. In this paper, we point out a mistake of their proof and give an improved proof.     

Acyclic 4-choosability of planar graphs

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. Given a list assignment L={L(v)∣v∈V} of G, we say G is acyclically L-list colorable if 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 planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.     

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.     

Edge choosability of planar graphs without 5-cycles with a chord

Let G be a plane graph having no 5-cycles with a chord. If either Δ≥6, or Δ=5 and G contains no 4-cycles with a chord or no 6-cycles with a chord, then G is edge-(Δ+1)-choosable, where Δ denotes the maximum degree of G.     

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.     

On 3-colorable planar graphs without prescribed cycles

In this paper we prove that every planar graph without cycles of length 4, 5, 6 and 8 is 3-colorable.     

Total colorings of planar graphs without adjacent triangles

Let G be a planar graph without adjacent 3-cycles, that is, two cycles of length 3 are not incident with a common edge. In this paper, it is proved that the total coloring conjecture is true for G; moreover, if Δ(G)≥9, then the total chromatic number χ^″(G) of G is Δ(G)+1. Some other related results are obtained, too.