Bounds on the Generalised Acyclic Chromatic Numbers of Bounded Degree Graphs

We give upper bounds for the generalised acyclic chromatic number and generalised acyclic edge chromatic number of graphs with maximum degree d, as a function of d. We also produce examples of graphs where these bounds are of the correct order. Part of this work supported by an Australian Research Council Postdoctoral Fellowship     

乘积图的全色数

本文得到了有关乘积图的全色数的一些结果,并利用这些结果证明了Mesh图和Tours-图均满足全色数猜想．特别,几乎所有的Mesh-图都是第一类图．     

Clique Chromatic Numbers of Intersection Graphs

Mathematical Notes -     

Chromatic Numbers of Some Distance Graphs

Mathematical Notes - For positive integers n > r > s, G(n, r, s) is the graph whose vertices are the r-element subsets of an n-element set, two subsets being adjacent if their...     

图的度序列

图的度序列是图论研究中一个重要的课题．至今已发表了４００余篇文章．本文概述这一课题的某些进展，其中包括了可图序列的判准、蕴含Ｐ可图序列和强迫Ｐ可图序列的一些主要结论，同时列出了一些有待进一步研究的问题．     

图中的度序列

给出了一个非减的非负整数序列是某个图的度序列的一个新刻划.     

Chromatic Numbers of Cayley Graphs on Z and Recurrence

Dedicated to the memory of Paul Erdős In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but never published, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. Received February 7, 2000     

The Harmonious Chromatic Number of Bounded Degree Graphs

A harmonious colouring of a simple graph G is a proper vertexcolouring such that each pair of colours appears together onat most one edge. The harmonious chromatic number h(G) is theleast number of colours in such a colouring. Let d be a fixed positive integer, and >0. We show that thereis a natural number M such that if G is any graph with mM edgesand maximum degree at most d, then the harmonious chromaticnumber h(G) satisfies      

某些中间图的邻点可区别E-全色数

设G（V,E）是简单连通图,T（G）为图G的所有顶点和边构成的集合,并设C是k-色集（k是正整数）,若T（G）到C的映射f满足：对任意uv∈E（G）,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,并且C（u）≠C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.那么称f为图G的邻点可区别E-全染色（简记为k-AVDETC）,并称χ_（at）~e（G）=min{k｜图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M（G）就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数.     

高度图的关联色数

为了解决强边着色猜想,1993年,Brualdi和Massey（Discrete Math. (122)51-58)引入了关联着色概念,陈东灵等证明了对于△（G）=n-2的图G.inc(G)≤△(G) 2,其中n是G的阶数,本将进一步探讨在什么条件下,它的关联色数肯定是△（G） 1,又在什么条件下,肯定是△（G） 2。     

Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

Acta Mathematicae Applicatae Sinica, English Series - Let G be a multigraph. Suppose that e = u1v1 and e′ = u2v2 are two edges of G. If e ≠ e′, then G(e, e′) is the graph...     

On Ramsey Numbers for Arbitrary Sequences of Graphs

Doklady Mathematics - In this work, we study nontrivial generalizations of Ramsey numbers to the case of arbitrary sequences of graphs. For many classes of sequences, exact values or asymptotics of...     

The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels

The notion of the star chromatic number of a graph is a generalization of the chromatic number. In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. The first two families are derived from a 3-or 5-wheel by subdivisions, their star chromatic numbers being 2+2/(2n + 1), 2+3/(3n + 1), and 2+3(3n−1), respectively. The third family of planar graphs are derived from n odd wheels by Hajos construction with star chromatic numbers 3 + 1/n, which is a generalization of one result of Gao et al. Received September 21, 1998, Accepted April 9, 2001.     

由轮图导出的某些平面图类的星色数

图的星色数是通常色数概念的推广.本文求出了几类由轮图导出的平面图的星色数.前两类是由3-或5-轮图经细分等构造出的,其星色数分别为2+2/(2n+1),2+3/(3n+1)和2+3/(3n-1).第三类平面图是由n-轮图经过Hajos构造得到的,其星色数为3+1/n.本类图的星色数结果推广了已有结论.     

An Extremal Problem on Degree Sequences of Graphs

 Let G=(I _n ,E) be the graph of the n-dimensional cube. Namely, I _n ={0,1} ⁿ and [x,y]∈E whenever ||x−y||₁=1. For A⊆I _n and x∈A define h _A (x) =#{y∈I _n A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I _n of size 2 ⁿ⁻¹ we have for a universal constant K independent of n. We prove a related lower bound for graphs: Let G=(V,E) be a graph with . Then , where d(x) is the degree of x. Equality occurs for the clique on k vertices. Received: January 7, 2000 RID="*" ID="*" Supported in part by BSF and by the Israeli academy of sciences     

On Size Ramsey Numbers of Graphs with Bounded Degree

G on n vertices with maximum degree three and the size Ramsey number where and c are positive constants. Received December 21, 1998     

Chromatic Numbers of Distance Graphs without Short Odd Cycles in Rational Spaces

Mathematical Notes - We prove the existence in rational spaces of distance graphs without short odd cycles whose chromatic number increases exponentially with the dimension of the space.     

On the Chromatic Number of Triangle-Free Graphs of Large Minimum Degree

We prove that, for each xed real number c > 1/3, the triangle-free graphs of minimum degree at least cn (where n is the number of vertices) have bounded chromatic number. This problem was raised by Erds and Simonovits in 1973 who pointed out that there is no such result for c < 1/3.