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.     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

图的Smarandachely邻点边色数的界

对图G的一个k-正常变染色法f,若图G中任意相邻两点的相邻边色集合互相不包含,那么称f为图G的一个k-Smarandachely邻点边染色(简记为k-SEC),而最小的正整数k称为图G的Smarandachely邻点边色数.尝试应用Lovasz局部引理来得到了Smarandachely邻点边色数的上界.     

Star Edge Coloring of Some Classes of Graphs

A star edge coloring of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this article, we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.     

联图的关联着色

图G的一个κ-关联着色是指从G的关联集I(G)到颜色集{1,2,…,κ}的一个映射,满足任意一对相邻的关联分配到不同的颜色.使得G有κ-关联着色的最小的数κ称为G的关联色数,记为X_i(G).研究了联图的关联着色,给出了G∨H的关联色数的一个上界,讨论了路与路,路与圈,圈与圈的联图的关联色数.     

一类3-正则图的邻强边染色

对简单图G(V,E),f是从V(G)∪E(G)到{1,2,…,k}的映射,k是自然数,若f满足(1)uv,uw∈E(G),u≠w,f(uv)≠f(uw);(2)uv∈E(G),C(u)≠C(v).则称f是G的一个邻强边染色,最小的k称为邻强边色数,其中C(u)={f(uv)|uv∈E(G)}.给出了一类3-正则重圈图的邻强边色数.     

圈的关联图的一般邻点可区别染色指标

给出了圈的关联图的一般邻点可区别色指标和一般邻点可区别全染色指标.     

Graphs of Large Girth with Prescribed Partial Circular Colourings

This paper completes the constructive proof of the following result: Suppose p/q2 is a rational number, A is a finite set and f₁,f₂,···,f_n are mappings from A to {0,1,···,p–1}. Then for any integer g, there is a graph G=(V,E) of girth at least g with such that G has exactly n (p,q)-colourings (up to equivalence) g₁,g₂,···,g_n, and each g_i is an extension of f_i. A probabilistic proof of this result was given in [8]. A constructive proof of the case p/q3 was given in [7].This research was partially supported by the National Science Council under grant NSC91-2115-M-110-004     

Acyclic List Edge Coloring of Graphs

A proper edge coloring of a graph is said to be acyclic if any cycle is colored with at least three colors. An edge-list L of a graph G is a mapping that assigns a finite set of positive integers to each edge of G. An acyclic edge coloring ? of G such that for any is called an acyclic L-edge coloring of G. A graph G is said to be acyclically k-edge choosable if it has an acyclic L‐edge coloring for any edge‐list L that satisfies for each edge e. The acyclic list chromatic index is the least integer k such that G is acyclically k‐edge choosable. We develop techniques to obtain bounds for the acyclic list chromatic indices of outerplanar graphs, subcubic graphs, and subdivisions of Halin graphs.     

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)=△.     

阿基米德图的面唯一极大染色

给定平面图G的一个正常κ-顶点染色φ:V(G)→{1,2,…,κ},若对G的每个面f,与f关联的顶点所染颜色的极大颜色在与f关联的顶点中仅出现一次,则称φ是图G的面唯一极大κ-染色.图G存在面唯一极大κ-染色的κ的最小值称为G的面唯一极大色数,记作χfum(G).本文研究了阿基米德图的面唯一极大色数,证得若图G是阿基米德图,则χfum(G)=4.     

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.     

若干联图的均匀全染色

图$G(V,E)$的全色数 $\chi_{t}(G)$就是将$V\bigcup E$分成彼此不相交的全独立分割集的最小个数。 如果任何两个$V\bigcup E$的全独立分割集的元素数目相差不超过1，那么 $V \bigcup E$的全独立分割集的最小个数就称为图$G$的均匀全色数，记为$\chi_{et}(G)$。 在本文中我们给出了当 $m \geq n \geq 3$ 时 $W_m\bigvee K_n$，$F_m \bigvee K_n$及$S_m \bigvee K_n$ 的均匀全色数.     

关于K_(2n)-E(C_m)的点可区别边色数

图G的一个k-正常边染色f被称为点可区别边染色是指任何两点的点及其关联边的色集合不同,所用最小的正整数k被称为G的点可区别边色数,记为X'_(vd)(G).用k_(2n)-E(C_m)表示2n阶完全图删去其中一条m阶路的边后得到的图,得到了K_(14)-E(C_4),K_(16)-E(C_4),K_(18)-E(C_5),K_(20)-E(C_5)的点可区别边色数分别为14,16,18,20.     

赋权图的圆染色与区间染色

赋权图的区间染色的定义与赋权图的圆染色的定义非常类型,唯一的区别就是将Ｇ的顶点对应圆周上的孤换为Ｇ的顶点对应区间上的子区间,讨论了赋权的圆染色与区染色的关系。     

若干图的点强全着色

图G(V,E)的一正常k-全着色σ称为G(V,E)的一个k-点强全着色,当且仅当v∈V(G),N[v]中的元素着不同颜色,其中N[v]={u|vu∈E(G)}∪{v}.并且vχsT(G)=min{k|存在G的一个k-点强全着色}称为G(V,E)的点强全色数.本文得到了一些特殊图的点强全色数χvTs(G),并提出猜想:对于简单图G,有k(G)≤χvTs(G)≤k(G)+1,这里k(G)表示图G中所有顶点间距离不超过2的点集的最大顶点数.     

一些联图的均匀全染色

对一个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.本文证明了图在若干情况下的均匀全色数定理,得到C_m∨S_nC_m∨ F_n和C∨W_n的均匀全色数.     

图的围长与无圈边色数之间的关系

对于一个图G的正常边着色，如果此种边着色使得该图没有2—色的圈，那么这种边着色被称为是G的无圈边着色．用d(G)表示图G的无圈边色数，即G的无圈边着色中所使用的最小颜色数．Alon N，Sadakov B and Zaks A在[1]中有如下结果：对于围长至少是2000△(G)log△(G)的图G，有d(G)≤△ 2，其中△是图G的最大度．我们改进了这个结果，得到了如下结论：对于围长至少是700△(G)log△(G)的图G，有d(G)≤△ 2．