Mycielski图的循环色数

通过引入一类点集划分的概念,研究了Mylielski图循环染色的性质,证明了当完全图的点数足够大时,它的Mycielski图的循环色数与其点色数相等.     

若干Mycielski图的均匀染色

如果图G的一个正常顶点染色满足任两个色类中的顶点数相差不超过1,则称为G的均匀染色.研究了一些Mycielski图的均匀染色,给出了路、圈、完全图和广义星图的Mycielski图的均匀色数.     

图M（Pn）和M（Gn）的点可区别均匀边染色

用构造法研究了路和圈的Mycielski图的点可区别均匀边染色,得到了路和圈的Mycielski图的点可区别均匀边色数,验证了它们满足点可区别均匀边染色猜想（VDEECC）．     

图M(S_n)和M(F_n)的点可区别均匀边色数

如果图G的一个正常边染色满足任意两个不同点的关联边色集不同,且任意两种颜色所染边数目相差不超过1,则称为点可区别均匀边染色(VDEEC),其所用最少染色数称为点可区别均匀边色数.本文用构造法研究了一些Mycielski图的点可区别均匀边染色,得到了星和扇的Mycielski图的点可区别均匀边色数,验证了它们满足点可区别均匀边染色猜想.     

图与其Mycielski图关联色数的关系

本文证明了对n阶图G,若其最大度△(G)的2倍不等于n,且G的关联色数等于△(G) 1,则M(G)的关联色数为△(M(G)) 1．同时还研究了树和完全二部图的Mycielski图的关联色数．文末提出了M(G)的关联色数猜想,其中M(G)为图G的Mycielski图．     

推广的Mycielski图和类推广的Mycielski图的(邻点可区别的)全染色

Mycielski图是在1955年由Mycielski首先提出的,推广的Mycielski图是在2003年由Peter Che Bor Lam,林文松等给出的Mycielski图的一个自然推广,且研究了它的圆色数.目前关于推广的Mycielski图性质以及它们在点色数,分数色数,圆色数等方面已有许多研究.本文定义了推广的Mycielski图的另一推广称为类推广的Mycielski图,且探讨了推广的Mycielski图和类推广的Mycielski图在全染色、邻点可区别全染色方面与原基础图的关系,从而也得到了它们满足全染色猜想和邻点可区别全染色猜想及它们达到全色数和邻点可区别的全色数的下界的一些充分条件.     

图M(P_m)和M(C_m)的点可区别边色数

本文研究了圈Cm和路Pm的Mycielski图的点可区别边染色问题.利用构造法给出了M(Cm)图的点可区别边染色法,得到了它的点可区别边色数,进而从图的结构关系,有效获得了M(Pm)图的相应点可区别边染色法和其边色数.该方法对研究存在结构关系的图染色问题具有重要的借鉴意义.     

关于完全图的Mycielski图的循环色数的若干结果

给出了任意图G的多重Myeielski图M^m(G)的简单定义方式，用不同的方法证明了当完全图Kn的阶数n足够大时，M^m(Kn)的循环色数等于其点色数．特别证明了，n=7，8，9时，M^3(Kn)的循环色数等于其点色数，从而使得“当n≥m 2，有xc(M^m))=x(M^m(Kn))=m n成立”的猜想有了更新的进展．     

平面图的循环色数及临界性

循环着色是普通着色的推广．本文中,我们研究了一类平面图-“花图”的循环着色问题,证明了由2r 1个长为2n 1的圈构成的“辐路”长度为m的花图Fr,m,n的循环色数是2 1/(n-m/2),并证明了在这类图中去掉任何一个点或边后,循环色数都严格减少但普通色数不减少,即这类图是循环色临界的但不是普通色临界的．同时,我们还研究了循环着色与图Gkd中的链之间的关系,给出了两个等价的条件．     

若干Mycielski图的点可区别均匀全色数

研究了一些Mycielski图的点可区别均匀全染色(VDETC),利用构造法给出了路、圈、星和扇的Mycielski图的点可区别均匀全色数,验证了它们满足点可区别均匀全染色猜想(VDETCC).     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

完全图的广义Mycielski图的邻强边染色

对|V(G)|≥3的连通图G,若κ-正常边染色法满足相邻点的色集合不相同,则称该染色法为κ-邻强边染色,其最小的κ称为图G的邻强边色数。张忠辅等学者猜想:对|V(G)|≥3的连通图G,G≠C_5其邻强边色数至多为△(G)+2,利用组合分析的方法给出了完全图的广义Mycielski图的邻强边色数,从而验证了图的邻强边染色猜想对于此类图成立。     

CIRCULAR CHROMATIC NUMBER AND MYCIELSKI GRAPHS

For a general graph G, M(G) denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number Xc(M(G)) equals to the chromatic number X(M(G)), which have improved some best sufficient conditions published up to date.     

On directed local chromatic number,shift graphs,and Borsuk‐like graphs

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of “topologically t‐chromatic” graphs. We show that this minimum for large enough t‐chromatic Schrijver graphs and t‐chromatic generalized Mycielski graphs of appropriate parameters is ?t/4?+1. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 65‐82, 2010     

推广的奇轮的圆色数

图G的圆色数(又称"星色数")xc(G)是Vince在1988年提出的,它是图的色数 的自然推广.本文由奇轮出发构造了一族平面图,并证明了此类图的圆色数恰恰介于2和 3之间,填补了该领域的空白.     

A new coloring theorem of Kneser graphs

In 1997, Johnson, Holroyd and Stahl conjectured that the circular chromatic number of the Kneser graphs KG(n,k) is equal to the chromatic number of these graphs. This was proved by Simonyi and Tardos (2006) [13] and independently by Meunier (2005) [10], if χ(KG(n,k)) is even. In this paper, we propose an alternative version of Kneser's coloring theorem to confirm the Johnson-Holroyd-Stahl conjecture.     

Vertex‐distinguishing edge colorings of graphs

We consider lower bounds on the the vertex‐distinguishing edge chromatic number of graphs and prove that these are compatible with a conjecture of Burris and Schelp 8 . We also find upper bounds on this number for certain regular graphs G of low degree and hence verify the conjecture for a reasonably large class of such graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 95–109, 2003     

The circular chromatic numbers of planar digraphs

The circular chromatic number is a refinement of the chromatic number of a graph. It has been established in [3,6,7] that there exists planar graphs with circular chromatic number r if and only if r is a rational in the set {1} ∪ [2,4]. Recently, Mohar, in [1,2] has extended the concept of the circular chromatic number to digraphs and it is interesting to ask what the corresponding result is for digraphs. In this article, we shall prove the new result that there exist planar digraphs with circular chromatic number r if and only if r is a rational in the interval [1,4]. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 14–26, 2007     

The circular chromatic number of series‐parallel graphs

In this article, we consider the circular chromatic number χ_c(G) of series‐parallel graphs G. It is well known that series‐parallel graphs have chromatic number at most 3. Hence, their circular chromatic numbers are at most 3. If a series‐parallel graph G contains a triangle, then both the chromatic number and the circular chromatic number of G are indeed equal to 3. We shall show that if a series‐parallel graph G has girth at least 2 ⌊(3k − 1)/2⌋, then χ_c(G) ≤ 4k/(2k − 1). The special case k = 2 of this result implies that a triangle free series‐parallel graph G has circular chromatic number at most 8/3. Therefore, the circular chromatic number of a series‐parallel graph (and of a K₄‐minor free graph) is either 3 or at most 8/3. This is in sharp contrast to recent results of Moser [5] and Zhu [14], which imply that the circular chromatic number of K₅‐minor free graphs are precisely all rational numbers in the interval [2, 4]. We shall also construct examples to demonstrate the sharpness of the bound given in this article. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 14–24, 2000