a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NP-hard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Be-and timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�-colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3-colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first non-trivial approximation result as a function of the maximum degree. This result can be generalized to�-colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�-function. 1

By the complexity of a graph we mean the minimum number of union andintersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of booleanfunctions. We prove some lower bounds on the complexity of explicitly given graphs.This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic frame. We also formulateseveral combinatorial problems whose solution would have intriguing consequences in computational complexity.

We consider generalized graph coloring and several other extremal problems in graph theory. In classical coloring theory, we color the vertices (resp. edges) of a graph requiring only that no two adjacent vertices (resp. incident edges) receive the same color. Here we consider both weakenings and strengthenings of those requirements. We also construct twisted hypercubes of small radius and find the domination number of the Kneser graph K(n, k) when n ≥ 3 4 k2 + k if k is even, and when n ≥ 3 4 k2 − k − 1 4 when k is odd. The path chromatic number χP (G) of a graph G is the least number of colors with which the vertices of G can be colored so that each color class induces a disjoint union of paths. We answer some questions of Weaver and West [31] by characterizing cartesian products of cycles with path chromatic number 2.

We consider the minimal number of AND and OR gates in monotone circuits for quadratic boolean functions, i.e. disjunctions of length-2 monomials. The single level conjecture claims that monotone single level circuits, i.e. circuits which have only one level of AND gates, for quadratic functions are not much larger than arbitrary monotone circuits. In this paper we disprove the conjecture: there are quadratic functions in n variables whose monotone circuits have linear size whereas their monotone single level circuits require ).

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In this paper, we investigate some basic properties of fractional powers. In this regard, we show that for any non-bipartite graph G and positive rational numbers 2q+1. Next, we study the power thickness of G,

Abstract. MPI process placement can play a deterministic role concerning the application performance. This is especially true with nowadays architecture (heterogenous, multicore with di erent level of caches, etc.). In this paper, we will describe a novel algorithm called TreeMatch that maps processes to resources in order to reduce the communication cost of the whole application. We have implemented this algorithm and will discuss its performance using simulation and on the NAS benchmarks. 1

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex of G sees all k colors on its closed neighborhood. In this note, we characterize all fall colorings of Kneser graphs of type KG(n, 2) for n ≥ 2 and study some fall colorings of KG(n, m) in some special cases and introduce some bounds for fall colorings of Kneser graphs.

We prove that for each natural number k, KG(2k+1, k) is b-continuous. Then, we introduce some special conditions for graphs to be b-continuous.