Abstract: For fixed integersp, q an edge coloring of a complete graphK is called a (p, q)-coloring if the edges of everyKp⊑K are colored with at leastq distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromaticKp subgraphs. Letf(n, p, q) be the minimum number of colors needed for a (p, q)-coloring ofKn. We use the Local Lemma to give a general upper bound forf. We determine for everyp the smallestq for whichf(n, p, q) is linear inn and the smallestq for whichf(n, p, q) is quadratic inn. We show that certain special cases of the problem closely relate to Turan type hypergraph problems introduced by Brown, Erdős and T. Sos. Other cases lead to problems concerning proper edge colorings of complete graphs.

Abstract: It is shown that for every 2 > 0 and n > n0(2), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1 − 1 √ 2 − 2)n edges of the same color, contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors.

Abstract: It is proved that a planar graph with maximum degree Δ ≥ 11 has total (vertex-edge) chromatic number $Delta; + 1. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 53–59, 1997

Abstract: Summary The harmonious chromatic number of a graph is the least number of colours in a vertex colouring such that each pair of colours appears on at most one edge The achromatic number of a graph is the greatest number of colours in a vertex colouring such that each pair of colours appears on at least one edge This paper is a survey of what is known about these two parameters, in particular we look at upper and lower bounds, special classes of graphs and complexity issues Introduction A short survey of harmonious colourings was given by Wilson [80] in 1990 Since then a number of new results have appeared, and the close relationship between harmonious chromatic number and achromatic number has been observed The purpose of this new survey is to outline what is known about these parameters, and suggest some open problems A more detailed summary of results on the achromatic number, with a rather different emphasis, can be found in the forthcoming survey by Hughes and MacGillivray [51] We begin with the definitions of the two parameters Definitions A harmonious colouring of a graph G is a proper vertex colouring of G such that, for any pair of colours, there is at most one edge of G whose endpoints are coloured with this pair of colours The harmonious chromatic number of G, denoted h(G), is the least number of colours in a harmonious colouring of G

Abstract: Graph coloring has several important applications inVLSI CAD. Since graph coloring is NP-complete, heuristics are used to approximate the optimum solution. But heuristic solutions are typically 10% off, and as much as100% off, the minimum coloring. This paper shows thatsince real-life graphs appear to be 1-perfect, one can indeed solve them exactly for a small overhead.

Abstract: Motivated by the problem of efficient routing in all-optical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an L-regular bipartite graph are already colored with αL different colors, then the rest of the edges can be colored using at most (1+α/2)L colors. We also show that this bound is tight by constructing instances in which (1+α/2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see.

Abstract: A mixed graphGπ contains both undirected edges and directed arcs. Ak-coloring ofGπ is an assignment to its vertices of integers not exceedingk (also called colors) so that the endvertices of an edge have different colors and the tail of any arc has a smaller color than its head. The chromatic number γπ(G) of a mixed graph is the smallestk such thatGπ admits ak-coloring. To the best of our knowledge it is studied here for the first time. We present bounds of γ(G), discuss algorithms to find this quantity for trees and general graphs, and report computational experience.

Abstract: The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 5 k 5 1 5 m, the subset graph S,(k, 1) is a bipartite graph whose vertices are the k- and 1-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that sq(Sm(k, 1)) = (&) and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. @ 1997 John Wiley & Sons, Inc.

Abstract: An L-list coloring of a graph G is a proper vertex coloring in which every vertex v gets a color from a list L(v) of allowed colors. G is called k-choosable if all lists L(v) have exactly k elements and if G is L-list colorable for all possible assignments of such lists. Verifying conjectures of Erdos, Rubin and Taylor it was shown during the last years that every planar graph is 5-choosable and that there are planar graphs which are not 4-choosable. The question whether there are 3-colorable planar graphs which are not 4-choosable remained unsolved. The smallest known example far a non-4-choosable planar graph has 75 vertices and is described by Gutner. In fact, this graph is also 3 colorable and answers the above question. In addition, we give a list assignment for this graph using 5 colors only in all of the lists together such that the graph is not List-colorable. © 1997 John Wiley & Sons, Inc.

Abstract: We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vertices occur. The key result on the polynomially solvable cases includes a good characterization theorem on the existence of an optimal coloring of a perfect graph where a given stable set is precolored with only one color. The key negative result states that the 3-colorability of a graph whose odd circuits go through two fixed vertices is NP-complete. The polynomial algorithms use Grotschel, Lovasz and Schrijver's algorithm for finding a maximum clique in a graph, but are otherwise purely combinatorial. c © 1997 John Wiley & Sons, Inc. J Graph Theory 25:

Abstract: This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k,d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to at most d-self-colored neighbors. First, (2,d) coloring is shown NP-complete for d <= 1, even for planar graphs, and (3,1) coloring is also shown NP-complete for planar graphs (while there exists a quadratic algorithm to (3,2)-color any planar graph). A reduction from ordinary vertex coloring then shows (X,d) coloring NP-complete for any X <= 3, d <= 0, as well as hardness of approximation results. Second, a generalization of Delta + 1 coloring defects is explored for graphs of maximum degree Delta. Based on a theorem of Lovasz, we obtain an O(Delta E) algorithm to (k, \1floor (Delta/k \rfloor) color any graph; this yields an O(E) algorithm to (2,1)-color 3-regular graphs, and (3,2)-color 6-regular graphs. The generalization of Delta + 1 coloring is used in turn to generalize the polynomial-time approximate 3- and k-coloring algorithms of Widgerson and Karger-Motwani-Sudan to allow defects. For approximate 3-coloring, we obtain an O(Delta E) time algorithm to $(\lceil({8n \over d})^{.5}\rceil,d)$ color, and a polynomial time algorithm to $(O((\frac{n} {d})^{.387}), d)$ color any 3-colorable graph.

Abstract: TheR-domatic number of a graph is the maximum number of colors that can be used to color the vertices of the graph so that all vertices of the graph have at least one vertex of each color within distanceR.In this paper the problem of determining theR-domatic number of then-cube,P(n,R), is considered. The valueP(6,1) =5 is settled, and a conjecture by Laborde thatP(n, 1)?n asntends to infinity is proved. Best known upper and lower bounds on theR-domatic number of then-cube are given forn?16 andR?7.

Abstract: In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove that there exists no polynomial approximation algorithm with ratio O(|V| 0.5-e ) for the OCCP problem restricted to bipartite and interval graphs, unless P = NP. Furthermore, we propose approximation algorithms with ratio O(|V| 0.5 ) for bipartite, interval and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O([V| 1-e ) for the OCCP problem restricted to split, chordal, permutation and comparability graphs, unless P = NP.

Abstract: Load balancing is the process of enhancing the performance of a distributed system through a redistribution of loads among the processors. In our earlier work a load balancing algorithm based on graph coloring for link-oriented structures was proposed and studied in detail. In this paper we modify the algorithm by introducing an important factor known as the damping factor, D. This factor is used to strike a balance between the runtime of the algorithm and the average response time. We observe that considerable improvement in system performance is possible with the application of the load balancing algorithm. The primary performance metric used is the average response time of the system. We express the analytical results in terms of upper and lower bounds on the average response time.

Abstract: The problem of the edge coloring partial k-tree into two partial p- and q-trees with p, q < k is considered An algorithm is provided to construct such a coloring with p + q = k Usefulness of this result in a Lagrangian decomposition framework to solve certain combinatorial optimization problems is discussed