简单完整正则平面图

对简单完整正则平面图的特性和结构进行了分析和讨论 ,找出了简单完整正则平面图的可能的种类 .此外 ,对各种简单完整正则平面图的色数进行了求解 ,并用不同的方法给出了各个简单完整正则平面图的作色方案 .     

Variations on the Theorem of Birkhoff – von Neumann and Extensions

 The theorem of Birkhoff – von Neumann concerns bistochastic matrices (i.e., matrices with nonnegative real entries such that all row sums and all column sums are equal to one). We consider here real matrices with entries unrestricted in sign and we extend the notion of permutation matrices (integral bistochastic matrices); some generalizations of the theorem are derived by using elementary properties of graph theory. Received: October 10, 2000 Final version received: April 11, 2002     

组合数学与数论间的一些交叉问题初探

我们证明了有限域上的一类方程组解的个数与图的顶点着色数有密切关系，而这又对许多着色问题的产生了许多应用。另外，我们也用图论的一些技巧解决了数论中一些问题。     

Graph Coloring with Adaptive Evolutionary Algorithms

This paper presents the results of an experimental investigation on solving graph coloring problems with Evolutionary Algorithms (EAs). After testing different algorithm variants we conclude that the best option is an asexual EA using order-based representation and an adaptation mechanism that periodically changes the fitness function during the evolution. This adaptive EA is general, using no domain specific knowledge, except, of course, from the decoder (fitness function). We compare this adaptive EA to a powerful traditional graph coloring technique DSatur and the Grouping Genetic Algorithm (GGA) on a wide range of problem instances with different size, topology and edge density. The results show that the adaptive EA is superior to the Grouping (GA) and outperforms DSatur on the hardest problem instances. Furthermore, it scales up better with the problem size than the other two algorithms and indicates a linear computational complexity.     

Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs

We present in this article an evolutionary procedure for solving general optimization problems. The procedure combines efficiently the mechanism of a simple descent method and of genetic algorithms. In order to explore the solution space properly, periods of optimization are interspersed with phases of interaction and diversification. An adaptation of this search principle to coloring problems in graphs is discussed. More precisely, given a graphG, we are interested in finding the largest induced subgraph ofG that can be colored with a fixed numberq of colors. Whenq is larger or equal to the chromatic number ofG, then the problem amounts to finding an ordinary coloring of the vertices ofG.     

图的路色数问题的NP－完全性

一个给定的图是否存在用ｒ种颜色的正常Ｐｋ着色？称该问题为图的（ｋ，ｒ）路色数问题．本文研究其算法复杂性，并得到以下结果：对于任意给定的ｋ，２≤ｋ≤∞，图的（ｋ，２）路色数问题及直径为２的图的（ｋ，３）路色数问题都是ＮＰ－完全的；对于任意给定的ｋ，２≤ｋ≤∞，平面图的（ｋ，３）路色数问题也是ＮＰ－完全的．     

An inequality for the group chromatic number of a graph

We give an inequality for the group chromatic number of a graph as an extension of Brooks’ Theorem. Moreover, we obtain a structural theorem for graphs satisfying the equality and discuss applications of the theorem.     

Partitioning series-parallel multigraphs intoν*-excluding edge covers

We prove that, for any given vertexν* in a series-parallel graph G, its edge set can be partitioned into k= min{k′(G) 1,δ(G)} subsets such that each subset covers all the vertices of G possibly except forν*, whereδ(G) is the minimum degree of G and k′(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

Pm×Kn的邻点可区别全色数

设G是简单图．设f是一个从V(G)∪E(G)到{1,2,…,k}的映射．对每个v∈V(G),令C_f(v)={f(v)}∪{f(vw)|w∈V(G),vw∈E(G)}．如果f是k-正常全染色,且对任意u,v∈V(G),uv∈E(G),有C_f(u)≠C_f(v),那么称f为图G的邻点可区别全染色(简称为k-AVDTC)．数x_(at)(G)=min{k|G有k-AVDTC}称为图G的邻点可区别全色数．本文给出路P_m和完全图K_n的Cartesion积的邻点可区别全色数．     

关于多重联图的均匀全染色

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．本文证明了关于多重联图的若干情况下的均匀全色数定理,得到了若干特殊多重联图的均匀全色数．