广义P$\grave{\bf o}$lya 组合计

文中引入了P-置换图的概念．作为置换群的指标多项式和函数等价类配置多项式的推广形式分别定义了P-置换图的容量指标多项式与色权多项式，并给出了递归公式和相关定理，由此建立了计算P-置换图的色权多项式的一般方法和P-置换图的色轨道多项式的表达公式．Pblya计数定理是这一公式当约束图是空图时的特例．最后给出了P-置换图的色权多项式的一些基本性质和两个计算实例．     

de Bruijn定理的推广

de Bruijn定理是一种重要的组合计数方法，本文以非常自然的方式推广了这种方法．p图是图G在其顶点上的置换群P作用下形成的轨道．文中引进了P-图，P-图的色容指标，P-图关于色置换群H的色权多项式以及色对称与全色对称图等概念，建立了色权多项式的计算公式和一系列的组合公式及性质．     

正则树的双变量色多项式研究

最近Klaus Dohmen等人提出新的双变量色多项式概念,对此,本文提出—个—般性的减边公式.通过反复运用该公式,可以方便求得任何简单图的双变量色多项式.由此减边公式,研究了一些特殊图和多分支图的双变量色多项式公式.本文还研究了由互不相连的多个子图都与某个顶点相连而成的图的双变量色多项式计算的删点公式以及简单图的双变量色多项式系数和问题.进而,本文提出—个新概念—正则树.利用这个减边公式,研究了正则树的双变量色多项式计算公式和—些性质,以及正则树整子图的双变量色多项式公式及其有关性质.     

图的色多项式系数之和问题的研究

本文给出了任何简单图G(V,E)的色多项式P(G，λ)=∑i=1^vαiλ^i系数之和的公式：∑i=1^vαi={0ε≠0 1ε=0;并进行了证明，从而为判别一个多项式不是图的色多项式提供了一个必要条件．同时也分别给出了树、2-树、圈、轮图和完全图的色多项式系数绝对值之和的表达式．最后证明了任何简单连通图的色多项式系数绝对值之和∑i=1^v|αi|与边ε成正比，且必满足2^v-1≤∑i=1^v|αi|≤пi=1^vi．     

若干图簇的伴随多项式的因式分解及色性分析

我们通过研究图的伴随多项式的因式分解,给出了证明非色唯一图的一种新方法,同时得到若干图簇的色等价图的结构定理.     

去掉一些连续弦的轮图的补图的色多项式

本文给出计算图的色多项式的新方法。特别的,对轮图中去掉一些连续弦后所得到的图的补图,给出了它的色多项式的计算公式。     

色多项式的显示公式

本文利用完全图K_n恰有k个分支S~((n))={K_i∶1≤i≤n}-因子个数N(K_n,k)及第二类Stirling数S(n,k)之间关系,导出图的色多项式的显示公式刻画,并给出几类色多项式及用Stirling数表示的完全i部图的色多项式的显式公式。     

图的色多项式的几个递推公式

本文给出下列图的色多项式的递推公式:删去图的一个二次或三次顶点;图的一边换成长为 k 的路;图 G 由 G_1和 G_2重迭一条路所组成,以及 Cm 多重图的边细分图的色多项式。     

广义Peterson图的着色问题研究

图的着色问题是图论的重要研究内容之一,利用广义的Pólya定理和结合一些代数方法研究了广义Peterson图在不同约束条件下的着色问题,并给出了四种不同约束条件下的色多项式.     

ω_(aδ)~ω∪bω_δ形图簇的伴随分解及其补图的色等价性

研究图的伴随分解及其补图的色等价性.采用伴随多项式的性质讨论图的伴随分解式,通过图的伴随分解式确定其补图的色性.证明了形图簇的伴随多项式的分解定理,从上述定理得到了这类图簇的补图的色等价性.结论通过图的伴随分解研究其补图的色等价性,是有效的途径与方法,从图的伴随分解式容易看出其补图的色等价图的结构规律.     

On the polynomial time computability of the circular-chromatic number for some superclasses of perfect graphs

A main result in combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). The circular-clique and circular-chromatic number are well-studied refinements of these graph parameters, and circular-perfect graphs form the corresponding superclass of perfect graphs. So far, it is unknown whether the (weighted) circular-clique and circular-chromatic number of a circular-perfect graph are computable in polynomial time. In this paper, we show the polynomial time computability of these two graph parameters for some super-classes of perfect graphs with the help of polyhedral arguments.     

On chromatic and flow polynomial unique graphs

It is known that the chromatic polynomial and flow polynomial of a graph are two important evaluations of its Tutte polynomial, both of which contain much information of the graph. Much research is done on graphs determined entirely by their chromatic polynomials and Tutte polynomials, respectively. Oxley asked which classes of graphs or matroids are determined by their chromatic and flow polynomials together. In this paper, we found several classes of graphs with this property. We first study which graphic parameters are determined by the flow polynomials. Then we study flow-unique graphs. Finally, we show that several classes of graphs, ladders, Möbius ladders and squares of n-cycle are determined by their chromatic polynomials and flow polynomials together. A direct consequence of our theorem is a result of de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomial, Graphs Comb. 20 (2004) 105-119] that these classes of graphs are Tutte polynomial unique.     

Computing clique and chromatic number of circular-perfect graphs in polynomial time

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel et al. in Combinatorica 1(2):169–197, 1981). Perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs; we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one. We consider a well-studied superclass of perfect graphs satisfying this property, the circular-perfect graphs, and show that for such graphs both clique and chromatic number are computable in polynomial time as well. In addition, we discuss the polynomial time computability of further graph parameters for certain subclasses of circular-perfect graphs. All the results strongly rely upon Lovász’s Theta function.     

Signed graph coloring

Coloring a signed graph by signed colors, one has a chromatic polynomial with the same enumerative and algebraic properties as for ordinary graphs. New phenomena are the interpretability only of odd arguments and the existence of a second chromatic polynomial counting zero-free colorings. The generalization to voltage graphs is outlined.     

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

On the reconstruction of graph invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U-polynomial, the universal edge elimination polynomial ξ and the colored versions of the latter two are reconstructible.We also present a method of reconstructing boolean graph invariants, or in other words, proving recognizability of graph properties (of colored or uncolored graphs), using first order logic.     

Approximations for chromatic polynomials

A sequence of finite graphs may be constructed from a given graph by a process of repeated amalgamation. Associated with such a sequence is a transfer matrix whose minimum polynomial gives a recursion for the chromatic polynomials of the graphs in the sequence. Taking the limit, a generalised “chromatic polynomial” for infinite graphs is obtained.     

Optimal acyclic edge colouring of grid like graphs

We determine the values of the acyclic chromatic index of a class of graphs referred to as d-dimensional partial tori. These are graphs which can be expressed as the cartesian product of d graphs each of which is an induced path or cycle. This class includes some known classes of graphs like d-dimensional meshes, hypercubes, tori, etc. Our estimates are exact except when the graph is a product of a path and a number of odd cycles, in which case the estimates differ by an additive factor of at most 1. Our results are also constructive and provide an optimal (or almost optimal) acyclic edge colouring in polynomial time.     

A generalization of chromatic polynomial of a graph subdivision

Considering the partitions of a set into nonempty subsets, we obtain an expression for the number of all partitions of a given type. The chromatic polynomial of a graph subdivision is generalized, considering two sets of colors, and a general explicit expression is obtained for this generalization. Using these results, we determine the generalized chromatic polynomial for the particular case of complete graph subdivision.     

若干图类的Wiener指数的极值

一个图的Wiener指数是指这个图中所有点对的距离和.Wiener指数在理论化学中有广泛应用. 本文刻画了给定顶点数及特定参数如色数或团数的图中Wiener指数达最小值的图, 同时也刻画了给定顶点数及团数的图中Wiener指数达最大值的图.