几类特殊Corona图的b-染色数与b-连续性研究

在一个图G的正常k染色中,如果每一个颜色类中都至少存在一个顶点,使得其在其它的k-1个颜色类中都至少有一个邻居,则称这样的正常k染色为b-染色.一个图G的b-染色数是最大的正整数k,使得用k种颜色能够对G进行b-染色,用b(G)来表示.如果对于任意的正整数k:χ(G)≤k≤b(G),用k种颜色可以对图G进行b-染色,则称图G是b-连续的.设G1与G2为任意图,称图G=G_1·G_2为图G_1与G_2的Corona图,其中G包含G_1的一个拷贝,包含G_2的|V(G_1)|个拷贝,且G_1的第i个顶点与G_2的第i个拷贝的所有顶点都邻接.研究了路图与路图、星形图以及轮图所构成的Corona图P_n·P_m、P_n·K_(1,m)以及P_n·W_(m+1)的m-度,b-染色数与b-连续性.     

几类积图的团染色数

设G=（VE）为简单图,V和E分别表示图的点集和边集．图G的一个k-团染色是指点集V到色集{1,2,…,k）的一个映射,使得G的每个至少含两个点的极大团都至少有两种颜色．分别给出了任意两个图的团色数与它们通过笛卡尔积、Kronecker积、强直积或字典积运算后得到的积图的团色数之间的关系．     

完全多部图K_(1*r,3*(k-2))的m数

如果对一个图G的每个顶点v,任给一个k-列表L(v),使得G要么没有正常列表染色,要么至少有两种正常列表染色,则称图G具有M(k)性质.定义图G的m数为使得图G具有M(k)性质的最小整数k,记为m(G).已有研究表明,当k=3,4时,图K_(1*r,3*(k-2))具有M(k)性质,且当r≥2时,m(K_(1*r,3*(k-2)))=k.本文将上述结论推广到每一个k,证明了对任意r∈N~+,k≥3,图K_(1*r,3*(k-2))具有M(k)性质,且当k≥4,r≥(k-2)时,m(K_(1*r,3*(k-2)))=k.此外,得到图K_(1,3,3,3)的m数为4,该图是图K_(1*r,3*(k-2))中r=1,k=5时的特殊情况,同时也是现有研究中尚未解决的一个问题.     

路和平方路笛卡尔乘积图的r-多彩着色（英文）

图G的r-多彩k-着色是图G的一个正常k-着色,并满足G中的每一个顶点的邻点的颜色数至少为这个顶点的度d(v)和r的最小值.使得图G具有r-多彩k-着色的最小整数k称为图G的r-多彩色数,用X_r(G)表示.本文研究了路和平方路的笛卡尔乘积图的r-多彩着色,得出了其r-多彩色数的确切值.     

高度平面图的邻点可区别全染色

图G 的邻点可区别全染色是G 的一个正常全染色, 使得每一对相邻顶点有不同的颜色集合. G的邻点可区别全色数χ_a′′ (G) 是使得G 有一个k- 邻点可区别全染色的最小颜色数k. 本文证明了: 若G 是满足最大度Δ(G) ≥ 11 的平面图, 则χ_a′′ (G) ≤ Δ(G) + 3.     

围长至少为4的平面图的邻点可区别边色数

图G的邻点可区别边染色是G的正常边染色,使得每一对相邻顶点有不同的颜色集合．G的邻点可区别边色数X＇a（G）是使得G有一个k-邻点可区别边染色的最小正整数七．本文证明了：若G是围长至少为4且最大度至少为6的平面图,则X＇a（G）≤△＋2．     

外平面图的有界染色

图G的k-有界染色是图G的一个最多有k个顶点染同一种颜色的顶点染色．图 G的k-有界染色数Xk(G)是指对G进行k-有界染色用的最少颜色数．本文给出了n个顶点的外平面图能用[n/k]种颜色k-有界染色的一些充分条件．     

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.     

不含5-圈和相邻6-圈的平面图的全染色

图的正常k-全染色是用k种颜色给图的顶点和边同时进行染色,使得相邻或者相关联的元素(顶点或边)染不同的染色.使得图G存在正常k-全染色的最小正整数k,称为图G的全色数,用χ″(G)表示.证明了若图G是最大度△≥6且不含5-圈和相邻6-圈的平面图,则χ″(G)=△+1.     

仙人掌图的邻点被扩展和可区别全染色

设G为简单图.G的全k-染色是指k种颜色1,2,…,k对图G的全体顶点及边的一个分配.设c是图G的一个全k-染色,任意的x∈V(G),称■为点x的扩展和,其中N(x)={y∈V(G)|xy∈E(G)}.称图G的全k-染色c为邻点被扩展和可区别(简记为NESD),如果w(x)≠w(y),其中xy∈E(G).使得图G存在NESD全k-染色的最小值k被称为图G的邻点被扩展和可区别全色数,简记为egndi_∑(G).本文利用数学归纳法探讨了仙人掌图的邻点被扩展和可区别全染色,并证明了这类图的邻点被扩展和可区别全色数不超过2.该结论说明Flandrin等人提出的NESDTC猜想对于仙人掌图是成立的.     

Clique-Coloring Circular-Arc Graphs

A clique-coloring of a graph is a coloring of its vertices such that no maximal clique of size at least two is monochromatic. A circular-arc graph is the intersection graph of a family of arcs in a circle. We show that every circular-arc graph is 3-clique-colorable. Moreover, we characterize which circular-arc graphs are 2-clique-colorable. Our proof is constructive and gives a polynomial-time algorithm to find an optimal clique-coloring of a given circular-arc graph.     

无爪图上团横贯数的界

设 G=(V,E) 为简单图,图 G 的每个至少有两个顶点的极大完全子图称为 G 的一个团. 一个顶点子集 S\subseteq V 称为图 G 的团横贯集, 如果 S 与 G 的所有团都相交,即对于 G 的任意的团 C 有 S\cap{V(C)}\neq\emptyset. 图 G 的团横贯数是图 G 的最小团横贯集所含顶点的数目,记为~${\large\tau}_{C}(G)$. 证明了棱柱图的补图(除5-圈外)、非奇圈的圆弧区间图和 Hex-连接图这三类无爪图的团横贯数不超过其阶数的一半.     

Clique-Coloring Claw-Free Graphs

A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacsó et al. (SIAM J Discrete Math 17:361–376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839–938, 2008) and the limitation of the degree 7 is necessary since the line graph of \(K_{6}\) is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number \(R(3,3)=6\). In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable (\(m\ge 3\)), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable.     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

有关循环图C(n;{1,k})的独立数的一些结果

令G=(V(G),E(G))是一个简单有限无向图.如果V(G)的子集S中任意两个顶点均不相邻,则S是图G的一个独立集.顶点独立集大小的最大值,称为图G的独立数,记作α(G).本文研究了循环图C(n;{1,k})的独立数问题,并给出了当k=2,3,4,5时的准确值.     

实现同一度序列的图的最大团数

图G中最大完全子图的阶数称为G的团效．ω(π)和γ(π)分别表示实现度序列π=(d_1,d_2,…,d_n)的图的最大团数和最小团数．Erds,Jacobson和Lehel开始考虑确定具有相同度序列π的图的可能的团数问题．他们证明了对于充分大的n,有ω(π)-γ(π)-n一2n~(2/3)．在本文中,我们首先估计了一类特殊可图序列的ω(π)之值,其次我们建立了一个估计任意可图序列π的ω(π)之值的算法．     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.