The Entire Coloring of Series-Parallel Graphs

The entire chromatic number X_(vef)(G) of a plane graph G is the minimal number of colors needed for coloring vertices, edges and faces of G such that no two adjacent or incident elements are of the same color. Let G be a series-parallel plane graph, that is, a plane graph which contains no subgraphs homeomorphic to K_(4-) It is proved in this paper that X_(vef)(G)≤max{8, △(G) 2} and X_(vef)(G)=△ 1 if G is 2-connected and △(G)≥6.     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

A SEVEN-COLOR THEOREM ON EDGE-FACE COLORING OF PLANE GRAPHS

1 IntroductionLet G be a plane graph with the vertex set V(G), the edge set E(G), the faCe set F(G),and the maximum degree A(G). The edge-face chromatic number X.I (G) of G is the ndnimumnunther of colors assigned to E(G) U F(G) such that aliy two adjacent or incident elements havedifferent colors. By the definition, X.,(G) 2 A(G) is trivial. In 1975, MelnikovI4J raised thefollowing conjecture.,Coniecture 1.1 For every plane graph G, X.J (G) 5 A(G) 3.The conjecture has been ton…     

List Vertex-arboricity of Planar Graphs without Intersecting 5-cycles

The vertex-arboricity a(G)of a graph G is the minimum number of colors required for a vertex coloring of G such that no cycle is monochromatic.The list vertex-arboricity al(G)is the list-coloring version of this concept.In this paper,we prove that every planar graph G without intersecting 5-cycles has al(G)≤2.This extends a result by Raspaud and Wang[On the vertex-arboricity of planar graphs,European J.Combin.29(2008),1064-1075]that every planar graph G without 5-cycles has a(G)≤2.     

Acyclic Edge Coloring of 1-planar Graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G.The acyclic chromatic index χ’α(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.It was conjectured that every simple graph G with maximum degree Δ has χ’_α(G) ≤Δ+2.A1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge.In this paper,we show that every 1-planar graph G without 4-cycles h...     

Rainb ow Vertex-connection Numb er of Ladder and M¨obius Ladder

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder Ln, we determine the exact value of srvc(Ln) for n even. For n odd, upper and lower bounds of srvc(Ln) are obtained. We also give upper and lower bounds of the (strong) rainbow vertex-connection number of M¨obius Ladder.     

Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has lc(G) = Δ(2G )+ 1 if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G s...     

Ladder图和Mbius Ladder图的彩虹点连通数（英文）

A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc(G). Observe that rvc(G) ≤ srvc(G) for any nontrivial connected graph G. In this paper, for a Ladder L_n,we determine the exact value of srvc(L_n) for n even. For n odd, upper and lower bounds of srvc(L_n) are obtained. We also give upper and lower bounds of the(strong) rainbow vertex-connection number of Mbius Ladder.     

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.     

The total chromatic number of regular graphs of high degree

The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) 32 |V (G)| + 263 , where d(G) denotes the degree of a vertex in G, then χT (G) d(G) + 2.     

系列平行图的边面染色

一个平面图G的边面色数xef(G)是指对G的边和面进行染色所用最少的颜色数目,并同时使得相邻或相关联的两个元素间染不同颜色.若G是一个系列平行图,也就是不含K_4的剖分作为子图的平面图,则有Xef(G)≤max{7,△(G) 1};同时如果G还是2-连通的且△(G)>6,则有Xef(G)=△.     

系列平行图的邻强边色数

本文研究了系列平行图的邻强边染色.从图的结构性质出发,利用双重归纳和换色的方法证明了对于△(G)=3,4的系列平行图满足邻强边染色猜想;对于△(G)≥5的系列平行图G, 有△(G)≤x'as(G)≤△(G) 1,且x'as(G)=△(G) 1当且仅当存在两个最大度点相邻,其中△(G)和x'as(G)分别表示图G的最大度和邻强边色数.     

广义图K(n,m)的全色数

1965年,M.Behzad和Vizing分别提出了著名的全着色猜想:即对于简单图G有:XT(G)≤△+2,其中△是图G的最大度.本文确定了完全图Kn的广义图K(n,m)的全色数,并利用它证明了Lm×Kn(m≥3)是第Ⅰ型的.     

大围长图的广义无圈染色

图的顶点染色称为是r-无圈的,如果它是正常染色,使得每一个圈C上顶点的颜色数至少为min{|C|,r}.图G的r-无圈染色数是图G的r-无圈染色中所用的最少的颜色数.我们证明了对于任意的r≥4,最大度为△、围长至少为2(r-1)△的图G的r-无圈染色数至多为6(r-1)△.     

关于图的L(3,2,1)-标号问题

图G的L( 2 ,1 )标号是一个从顶点集V(G)到非负整数集的函数f(x) .使得若d(x ,y) =1 .则｜f(x) -f(y) ｜≥ 2 ;若d(x ,y) =2 ,则｜f(x) -f(y)｜≥ 1 .图G的L( 2 ,1 )标号数λ(G)是使得G有max{f(v) ∶v∈V(G) }=k的L( 2 ,1 )标号中的最小数k .本文将L( 2 ,1 ) 标号问题推广到更一般的情形即L( 3,2 ,1 ) 标号问题 .我们首先定义了图G的顶点 3 着色及图的 3 色数 χ3 (G)等有关概念 ,并推导出 3 色数 χ3 (G)的上界 ;然后根据 χ3 (G)与λ3 (G)的关系 ,得出了对一般图G ,有λ3 (G) ≤ 3maxH Gδ(H) (Δ2 -Δ 1 )这一一般关系式 ;最后证明了对一般平面图G ,有λ3 (G)≤ 1 5(Δ2 -Δ 1 ) ,并得出了其它几类平面图的λ3 (G)的上界 .     

具有较小直径的三类图的对极色数

对一个连通图G,令d(u,v)表示G中两个顶点间u和v之间的距离,d表示G的直径.G的一个对极染色指的是从G的顶点集到正整数集(颜色集)的一个映射c,使得对G的任意两个不同的顶点u和v满足d(u,v)+|c(u)-c(v)|≥d.由c映射到G的顶点的最大颜色称为c的值,记作ac(c),而对G的所有对极染色c,ac(c)的最小值称为G的对极色数,记作ac(G).本文确定了轮图、齿轮图以及双星图三类图的对极色数,这些图都具有较小的直径d.     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

最大度不小于5的外平面图的邻强边染色

图G(V,E)的一k-正常边染色叫做k-邻强边染色当且仅当对任意uv∈E(G)有,f[u]≠f[v],其中f[u]={f(uw)|uw∈E(G)},f(uw)表示边uw的染色.并且x'as(G)=min{k|存在k-图G的邻强边染色}叫做图G的图的邻强边色数.本文证明了对最大度不小于5的外平面图有△≤x'as(G)≤△ 1,且x'as(G)=△ 1当且仅当存在相邻的最大度点.     

△-匹配与边面全色数

设Ｇ为　（Ｇ）≥５的外平面图且　　（Ｇ）为Ｇ的边面全色数。本文证明了：且当且仅当Ｇ含有一个由内边组成且覆盖Ｇ的每一个最大度点的匹配。     

若干图的强染色

图 G(V,E)的一正常 k-染色 σ称为 G(V,E)的 - k-强染色当且仅当对任何两个不同顶点 u和 v,只要d(u,v)≤ 2 ,则 u、v染不同颜色 (这里 d(u,v)表示 u,v之间的距离 ) ,并称 xs(G) =min{ k|存在 G的 - k-强染色 }为 G的强色数 ,本文得到 θ-图 ,Cm,n图 ,Halin图的强色数 xs(G)