关于Sm∨Sn的边色数和邻强边色数

本文研究了m＋1阶的星Sm和n＋1阶的星Sn的联图Sm∨Sn的边染色和邻强边染色．得到了Sm∨Sn的边色数和邻强边色数。     

图K_3~cVK_t的点可区别正常边染色

图G的正常边染色称为是点可区别的,如果对G的任意两顶点的关联边的颜色构成的集合不同.对图G进行点可区别正常边染色所需要的最少颜色数称为图G的点可区别正常边色数,记为x_s'(G).给出了3阶空图与t阶完全图的联图的点可区别正常边色数.     

Sm ∨ Kn,n的边色数和均匀全色数

本文得到了m 1阶星和完全等二部图联图的边色数和均匀全色数.     

关于图是(r,n)-临界图的一个邻域条件

设n，m和r是满足r≥2，n≥0，m≥3的整数，且当r是奇数时，假设r≥m-1．称一个图为K1,m-free，如果它不包含以Kt,m为导出的子图．称一个图G为一个（r,n）-临界图，如果在删去G的任意n个点后，剩下G的子图都有一个r-因子,设G是一个Kl,m-free的（n＋1）-连通图，且阶为｜G｜以及r（｜G｜≥n）是偶数,证明了：如果G的最小度至少是r＋n＋m-1，阶｜G｜≥8r5＋n，并且对V（G）的任意独立点集{x1，x2}都有｜NG（x1）∪NG（x2）｜≥（｜G｜＋n）／2，那么G是一个（r,n）-临界图．关于G的最小度和｜NG（x1）∪NG（X2）｜的下界是紧的。     

色指数临界图的一些新结论

图的边色数是指对图的边进行染色使得任意两相邻边染不同的颜色所需要的最少的颜色数.1965年,Vizing证明了任意最大度为△的图的边色数或者是△或者是△+1.若G是连通的,且G的每一条边e均有X′(G-e)相似文献   

一些图的Mycielski图的均匀邻强边染色

如果图G的一个正常边染色满足相邻点的色集不同,且任意两种颜色所染边数目相差不超过1,则称为均匀邻强边染色,其所用最少染色数称为均匀邻强边色数.本文得到了路、圈、星和扇的Mycielski图的均匀邻强边色数.     

路与星联图的均匀邻强边色数

如果图G的一个正常边染色满足相邻点的色集不同,且任意两种颜色所染边数相差不超过1,则称为均匀邻强边染色,其所用最少染色数称为均匀邻强边色数.本文得到在m=1,2,3,n≥1和m=n≥4时的均匀邻强边色数.     

近完全图的邻点可区别正常边色数

引入了近完全图的概念,并根据其结构特征,给出了近完全图的邻点可区别正常边色数.该结果揭示了完全图中删去一个匹配后其邻点可区别正常边色数的变化情况.     

图K(r,2)的邻强边色数

本文给出了每部有2个点的完全r-部图(r≥2)的邻强边色数.     

若干图的倍图的均匀邻强边染色

如果图G的一个正常边染色满足相邻点的色集不同,且任意两种颜色所染边数目相差不超过1,则称为均匀邻强边染色,其所用最少染色数称为均匀邻强边色数．本文得到了星、扇和轮的倍图的均匀邻强边色数．     

关于图的3-条件色数

For integers k0,r0,a(k,r)-coloring of a graph G is a proper k-coloring of the vertices such that every vertex of degree d is adjacent to vertices with at least min{d,r}diferent colors.The r-hued chromatic number,denoted byχr(G),is the smallest integer k for which a graph G has a(k,r)-coloring.Define a graph G is r-normal,ifχr(G)=χ(G).In this paper,we present two sufcient conditions for a graph to be 3-normal,and the best upper bound of 3-hued chromatic number of a certain families of graphs.     

Orientations of graphs with uncountable chromatic number

Motivated by an old conjecture of P. Erdős and V. Neumann‐Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that, consistently, there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected case: any graph with uncountable chromatic number contains a 4‐cycle. Next, we prove that several well‐known graphs (uncountable complete graphs, certain comparability graphs, and shift graphs) admit orientations with uncountable dichromatic number in ZFC. However, we show that the statement “every graph G of size and chromatic number ω₁ has an orientation D with uncountable dichromatic number” is independent of ZFC. We end the article with several open problems.     

On chromatic‐choosable graphs

A graph is chromatic‐choosable if its choice number coincides with its chromatic number. It is shown in this article that, for any graph G, if we join a sufficiently large complete graph to G, then we obtain a chromatic‐choosable graph. As a consequence, if the chromatic number of a graph G is close enough to the number of vertices in G, then G is chromatic‐choosable. We also propose a conjecture related to this fact. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 130–135, 2002     

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.     

A Characterization of b‐Perfect Graphs

A b‐coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b‐chromatic number of a graph G is the largest integer k such that G admits a b‐coloring with k colors. A graph is b‐perfect if the b‐chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b‐perfect if and only if it does not contain as an induced subgraph a member of a certain list of 22 graphs. This entails the existence of a polynomial‐time recognition algorithm and of a polynomial‐time algorithm for coloring exactly the vertices of every b‐perfect graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:95–122, 2012     

色数等于选择数的4-正则图

设Hn(n≥5)表示一个图:以1,2,．．．,n为顶点,两个点i和j是相邻的当且仅当|i-j|≤2,其中加法取模n．这篇文章证明了,Hn的色数等于它的选择数．结果被用于刻画最大度至多2的图的列表全色数．     

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)△.     

3-连通、高次和坚韧图周长的估计(Ⅰ)

设G是一个n阶3-连通图,周长为C(G),独立数为,若G是1-坚韧的,且,则G的每一个最长圈是控制圈且;又若G是5/3-坚韧的或,则G是Hamilton图。