Abstract: An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to \({\cal O}\)(n/(log n)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems.

Abstract: Online graph coloring, in which the vertices are presented one at a time, is considered. Each vertex must be assigned a color, different from the colors of its neighbors, before the next vertex is given. The class of d-inductive graphs is treated. A graph G is said to be d-inductive if the vertices of G can be numbered so that each vertex has at most d edges to higher numbered vertices. First Fit (FF) is the algorithm that assigns each vertex the lowest numbered color possible. It is shown that if G is d-inductive, then FF uses O(d log n) colors on G. This yields an upper bound of O(log n) on the performance ratio of FF on chordal and planar graphs. FF does as well as any online algorithm for d-inductive graphs; it is shown that for any d and any online graph-coloring algorithm A, there is a d-inductive graph that forces A to use Omega (d log n) colors to color G. Online graph coloring with lookahead is also investigated. >

Abstract: Approximate graph coloring takes as input a graph and returns a legal coloring which is not necessarily optimal. We improve the performance guarantee, or worst-case ratio between the number of colors used and the minimum number of colors possible, toO(n(log logn)3/(logn)3), anO(logn/log logn) factor better than the previous best-known result.

Abstract: It is shown that randomization helps in coloring graphs online, and a simple randomized online algorithm is presented. For 3-colorable graphs the expected number of colors the algorithm uses is O((n log n)/sup 1/2/). The algorithm runs in polynomial time and compares well with the best known polynomial-time offline algorithms. A lower bound is proved for the randomized algorithm. >

Abstract: When can a k-edge-coloring of a subgraph K of a graph G be extended to a k-edge-coloring of G? One necessary condition is that for all X ⊆ E(G) - E(K), where μi(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends.

Abstract: A graph is called Class 1 if the chromatic index equals the maximum degree. We prove sufficient conditions for simple graphs to be Class 1. Using these conditions we improve results on some edge-coloring theorems of Chetwynd and Hilton. We also improve a theorem concerning the 1-factorization of regular graphs of high degree.

Abstract: This paper is concerned with the design and performance of coloration neighborhood search (CNS) algorithms for the general graph coloring problem. A coloration neighborhood is an implicit mapping F : II --+ 25 where H is the set of all colorations of a graph, G = (V,E). We require that F(C) be easily enumerable and, ideally, we would like to be able to choose a coloration from r(C) uniformly at random. CNS algorithms traverse a neighborhood structure in search of a “good” coloration by repeatedly sampling the neighborhood of the current coloration as shown in Figure 1, They algorithms are computation intensive, but produce better colorations than those obtained by doing multiple runs, for the same length of time, using randomized versions of the known polynomial time approximation algorithms [3,8]. CNS algorithms are fully specified by giving a search framework (selection of C, acceptance of C and a termination criterion) as well as a coloration neighborhood. Well known frameworks include simulated annealing [3] and the tabu search method [23. Acceptance of a neighboring coloration is determined using an objective function that prescribes a value to each coloration. Smaller objective function values correspond to better colorations. However, using objective function improvement as the only criterion for acceptance is too strict; this approach often results in the search becoming trapped in a local optimum that is far away from a global optimum. One way to avoid this local trapping is to allow for the possibility of accepting a neighboring coloration that represents a move contrary to the direction of optimality-such a move is called a disimprovement. Search frameworks have a parametrized mechanism for accepting disimprovements that control the frequency of such moves and the magnitude of change to the objective function. A well known

Abstract: A linear forest is a forest in which each connected component is a path. The linear arboricity la(G) of a graph G is the least number of linear forests required to cover the edges of G. The linear arboricity conjecture is equivalent to the assertion that for every r-regular graph G

Abstract: Consider families of arcs on a circle. The minimum coloring problem on arc families has been shown to be NP-hard by Garey, Johnson, Miller and Papadimitriou. It is easy to show that 2q colors are sufficient for any arc family F, where q is the size of a maximum clique in F and 3q/2 colors are necessary for some families. It has long been open problem to find a coloring algorithm which uses no more than α·q colors , where α is strictly less than 2. In this paper we present such an algorithm with α=5/3. Our algorithm is based on: (1) an extension of an earlier result of Tucker on coloring special families and (2) a characterization of the existence of perfect matching in bipartite graphs. 1 Department of Computer Science, National Tsing-Hua University, Republic of China. 2 Institute of Information Sciences, Academia Sinica, Republic of China.

Abstract: Motivated by a question in cellular telecommunication technology, we investigate a family of graph coloring problems where several colors can be assigned to each vertex and no two colors are the same within any ball of radiusR. We find bounds and coloring algorithms for different kinds of graphs including trees,n-cycles, hypercubes and lattices. We briefly examine connections to Heawood's map color theorem and state a few conjectures and open problems.

Abstract: This paper introduces a new type of edge-coloring of multigraphs, called anfg-coloring, in which each color appears at each vertexv no more thanf(v) times and at each set of multiple edges joining verticesv andw no more thang(vw) times. The minimum number of colors needed tofg-color a multigraphG is called thefg-chromatic index ofG. Various upper bounds are given on thefg-chromatic index. One of them is a generalization of Vizing's bound for the ordinary chromatic index. Our proof is constructive, and immediately yields a polynomial-time algorithm tofg-color a given multigraph using colors no more than twice thefg-chromatic index.

Abstract: The efficiency of a parallel implementation of the conjugate gradient method preconditioned by an incomplete Cholesky factorization can vary dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we compare the performance of the preconditioned conjugate gradient method using these coloring orderings with a number of standard orderings on matrices arising from applications in structural engineering. Because optimal colorings for these systems may not be a priori known: we employ several graph coloring heuristics to obtain consistent colorings. Based on lower bounds obtained from the local structure of these systems, we find that the colorings determined by these heuristics are nearly optimal. For these problems, we find that the increase in parallelism afforded by the coloring-based orderings more than offsets any increase in the number of iterations required for the convergence of the conjugate gradient algorithm.

Abstract: In this paper we study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by Lov&z, Saks and Trotter [5]. Their algorithm colors graphs of chrs matic number x with no more than ((2xn)/log*n) colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For 3chromatic graphs the number of colors used, for example, is O( n log log log n/ log log n). We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors 0(2X~~ns(log n) x-1). For 3-colorable graphs the expected number of colors our algorithm uses is O(d*). All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time ofline algorithms. For instance, the best polynomial time algorithm known for 3-colorable graphs, due to Avrim Blum, uses O(n3/" poly-log(n)) colors [3]. In section 3 we prove a lower bound of a(&( 12]~~~l))X-1) for the randomized model. No lower bound for the randomized model was previously known. Our result improves even the best known lower bound for the deterministic case: a(( ,$:Gn)X-') for

Abstract: Several tools for use in approximation algorithms to color 3-chromatic graphs are presented. The techniques are used in an algorithm that colors any 3-chromatic graph with O(n/sup 3/8/)+O(n/sup 3/8+O(1)/) colors (or more precisely) O(n/sup 3/8/log/sup 5/8/ n) colors, which improves the previous best bound of O(n/sup 0.4+0(1)/) colors. The techniques are illustrated by considering a problem in which the 3-chromatic graph is created not by a worst-case adversary, but by an adversary each of whose decisions (whether or not to include an edge) is reversed with some small probability or noise rate p. This type of adversary is equivalent to the semirandom source of M. Santha and U.V. Vazirani (1986). An algorithm that will actually 3-color such a graph with high probability even for quite low noise rates (p>or=n/sup -1/2+ epsilon / for constant epsilon >0), is presented. >

Abstract: Let G=(V, E) be a strongly connected, aperiodic, directed graph having outdegree 2 at each vertex. A red-blue coloring of G is a coloring of the edges with the colors red and blue such that each vertex has one red edge and one blue edge leaving it. Given such a coloring, we define R:V→V by R(v)=w if there is a red edge from v to w. Similarly we define B:V→V. G is said to be collapsible if some composition of R's and B's maps V to a single vertex. The road coloring problem is to determine whether G has a collapsible coloring. The adjacency matrix, A, of G has a positive left eigenvector w=(w(v 1 ),..., w(v n )) with eigenvalue 2. We can assume that w's components are integers with no common factor. We call w(v) the weight of v. Let W≡Σ v∈V w(v), defined to be the weight of the graph. We prove that if G has a simple cycle of length relatively prime to W, then G is collapsibly colorable

Abstract: A theorem of Brooks guarantees that a maximum-degree-D graph can be properly colored with D colors if the graph does not contain a large complete subgraph. It is proved that finding such a coloring is in NC.

Abstract: Heuristics for Graph Coloring. Some sequential coloring techniques are reviewed. A few general principles for designing heuristics are outlined and recent coloring techniques baaed on tabu search are discussed.

Abstract: Problems and results are presented concerning the strong chromatic index, where the strong chromatic index is the smallest k such that the edges of the graph can be k-colored with the property that each color class is an induced matching. This parameter was suggested by Erdos and Nesetril and it is related to an extremal problem of Bermond, Bond and Peyrat concerning induced matchings of graphs.