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.     

Acyclic 5-choosability of planar graphs without 4-cycles

The conjecture on the acyclic 5-choosability of planar graphs (Borodin et al., 2002) as yet has been verified only for several restricted classes of graphs: those of girth at least 5 (Montassier, Ochem, and Raspaud, 2006), without 4- and 5-cycles or without 4- and 6-cycles (Montassier, Raspaud, and Wang, 2007), with neither 4-cycles nor chordal 6-cycles (Zhang and Xu, 2009), with neither 4- cycles nor two 3-cycles at distance less than 3 (Chen and Wang, 2008), and with neither 4-cycles nor intersecting 3-cycles (Chen and Raspaud, 2010). Wang and Chen (2009) proved that the planar graphs without 4-cycles are acyclically 6-choosable. We prove that a planar graph without 4-cycles is acyclically 5-choosable, which is a common strengthening of all above-mentioned results.     

不含4-圈和5-圈的平面图的非正常2-染色的一个新结果

设d_1,d_2,···,d_k是k个非负整数,若图G=(V,E)的顶点集V能被剖分成k个子集V_1,V_2,···,V_k,使得对任意的i=1,···,k,V_i的点导出子图G[Vi]的最大度至多为di,则称图G是(d_1,d_2,···,d_k)-可染的,本文证明了既不含4-圈又不含5-圈的平面图是(9,9)-可染的.     

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.     

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].     

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $V_i?\{v\in V\left(G\right):c\left(v\right)=i\}$ for $i=1$ and $2.$ We say that $G$ is $(d_1,d_2)$-colorable if $G$ has a 2-coloring such that $V_i$ is an empty set or the induced subgraph $G\left[V_i\right]$ has the maximum degree at most $d_i$ for $i=1$ and $2.$ Let $G$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for every positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $k.$ In contrast, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4),(3,5),$ and $(2,9).$     

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.     

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 6-choosability of Planar Graphs without 5-cycles and Adjacent 4-cycles

A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors. A graph G is acyclically k-choosable if for any list assignment L = {L(v) : v ∈ V(G)} with |L(v)| ≥ k for all v ∈ V(G), there exists a proper acyclic vertex coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V(G). In this paper, we prove that if G is a planar graph and contains no 5-cycles and no adjacent 4-cycles, then G is acyclically 6-choosable.     

On total chromatic number of planar graphs without 4-cycles

Let G be a simple graph with maximum degree A(G) and total chromatic number Xve(G). Vizing conjectured thatΔ(G) 1≤Xve(G)≤Δ(G) 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs isΔ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then Xve(G)≤8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.     

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.     

不含5-圈和6-圈的平面图的(2,1)-全标号

图G的(2,1)-全标号是对图G的顶点和边的一个标号分配,使得:(1)任意两个相邻顶点标号不同;(2)任意两条相邻边标号不同;(3)任意顶点与其相关联的边标号至少相差2.两个标号的最大差值称为跨度,图G的所有(2,1)-全标号的最小跨度称为(2,1)-全标号数,记为λ_2~T(G).本文证明了如果G是一个?=p+5的平面图,且G不包含5-圈和6-圈,那么λ_2~T(G)=2?-p,p=1,2,3.     

Structural properties and edge choosability of planar graphs without 4-cycles

Some structural properties of planar graphs without 4-cycles are investigated. By the structural properties, it is proved that every planar graph G without 4-cycles is edge-(Δ(G)+1)-choosable, which perfects the result given by Zhang and Wu: If G is a planar graph without 4-cycles, then G is edge-t-choosable, where t=7 if Δ(G)=5, and otherwise t=Δ(G)+1.     

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.     

On the 3-colorability of planar graphs without 4-, 7- and 9-cycles

In this paper, we mainly prove that planar graphs without 4-, 7- and 9-cycles are 3-colorable.