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Topic: Graph coloring problem

Related Topics

Cocoloring

Glossary of graph theory

Planar graph

Vertex cover problem

Clique problem

Traveling salesman problem

Project management

PRINCE2

	Graph theory - Open Encyclopedia (Site not responding. Last check: 2007-10-20)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs).
		Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
	open-encyclopedia.com /Graph_theory (881 words)

	Map / Graph Vertex Coloring Problems Using CCM
		Map coloring problem is a constraint satisfaction problem (CSP) to color the areas in a map using predefined number of colors so that the neighboring areas are in different colors.
		Graph vertex coloring problem is a CSP to color the vertices of a graph using predefined number of colors so that the vertices that are connected by an edge are in different colors.
		A problem of coloring a map on a plane is used as an example because it is suited for demonstration.
	www.ff.iij4u.or.jp /~kanada/ccm/coloring (696 words)

	e-Merge-ANT: Scheduling using Graph Coloring
		In graph coloring, a commonly used characteristic of a graph is its chromatic number which is the fewest number of colors required to color the graph so that no two nodes of the same color are connected by an edge.
		If we try to color a graph with fewer colors, we inevitably end up with conflicts (i.e., nodes of the same color connected by an edge): the equivalent situation in the scheduling problem occurs when the revisit period is too short compared with the dwell time.
		More generally, the number of colors compared with the chromatic number gives a measure of the difficulty of the coloring problem: if the number of colors is less than or greater than the chromatic number, the coloring problem is over constrained or under constrained, respectively.
	ants.kestrel.edu /challenge-problem/Y1/index.html (1152 words)

	[No title]
		Given a graph G(V, E), the goal of this NP-complete problem is to find a color assignment to every vertex in II such that arty pair of adjacent vertices must not receive the same color but also the total number of colors should be minimized.
		Graph coloring with adaptive evolutionary algorithms Eiben AE, Van der Hauw JK, Van Hemert JIJOURNAL OF HEURISTICS 4 (1): 25-46 JUN 1998 This paper presents the results of an experimental investigation on solving graph coloring problems with Evolutionary Algorithms (EAs).
		The problem is formulated as a restricted vertex colouring problem and a branch and bound algorithm is presented.
	mat.gsia.cmu.edu /COLOR03/BIB/Elsevier-web-of-science.doc (8405 words)

	The Artificial Unger Graph Coloring Applet (Site not responding. Last check: 2007-10-20)
		The graph coloring problem is an extension of the map coloring problem.
		However, in the graph coloring problem the graph is not necessarily planar nor bi-directional.
		To solve the general graph coloring problem with either algorithm, the applet starts with a color palette of n, where n is the number of vertices in the graph.
	www.duke.edu /~jmu2/color/gc.html (1773 words)

	Describe (Site not responding. Last check: 2007-10-20)
		Graph coloring problems may seem like puzzles, but they have many applications in industry, including scheduling, testing circuit boards, and assigning frequencies for radio and phone communications.
		The problem is to find a way to color the graph so that each incompatible pair is colored different colors, ideally using the fewest possible colors.
		There is a theorem that states that the minimum number of colors needed to color a graph so that two nodes that are linked by an edge are different colors is greater than or equal to the size of the largest clique.
	drwho.ee.ethz.ch /sepp/ilog-2003.11-as/doc/solver60/userman/graph2.html (1010 words)

	Graph Coloring
		Graph coloring is an interesting application to look at for several reasons.
		In addition, a large number of practical problems can be formulated in terms of coloring a graph, including many scheduling problems (Gondran and Minoux, 1984).
		Figure 45 is a simple variant (with solution) of the first graph coloring problem in the introduction: find a graph coloring that uses four colors such that red is used as often as possible.
	crl.nmsu.edu /users/sb/papers/thesis/node50.html (1301 words)

	1419 -- Graph Coloring (Site not responding. Last check: 2007-10-20)
		Colors are applied to the nodes of the graph and the only available colors are fl and white.
		The coloring of the graph is called optimal if a maximum of nodes is fl.
		The coloring is restricted by the rule that no two connected nodes may be fl.
	acm.pku.edu.cn /JudgeOnline/showproblem?problem_id=1419 (165 words)

	Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring - ...
		Finding challenging benchmarks for the maximum independent set problem (or equivalently, the minimum vertex cover problem) is not only of significance for experimentally evaluating the algorithms of solving this problem but also of interest to the theoretical computer science community.
		Given a graph G, a clique is a subset S of vertices in G such that each pair of vertices in S is connected by an edge.
		Since a clique is an independent set in the complementary graph, the maximum independent set problem and the maximum clique problem (which is one of the first shown to be NP-hard and has been extensively studied in graph theory and combinatorial optimization) are essentially equivalent.
	www.nlsde.buaa.edu.cn /~kexu/benchmarks/graph-benchmarks.htm (1534 words)

	Joseph Culberson's Graph Coloring Resources Page
		The main emphasis is on vertex coloring, and in particular on algorithms for obtaining vertex colorings.
		Over views of the four color theorem: A history and a discussion and New four color proof by Robertson, Sanders, Seymour and Thomas.
		Graph coloring is a restricted class of the constraint satisfaction problem.
	www.cs.ualberta.ca /%7Ejoe/Coloring (689 words)

	Distributional Graph Edge Coloring
		Venkatesan and Levin [VL88] constructed a randomized reduction from the distributional tiling problem to the distributional graph edge coloring problem.
		We distinguish the tiles by coloring the corners of the tiles.
		When a self-loop is colored, it is convenient to simply say that the looped node is colored (with the same color of the self-loop).
	www.uncg.edu /mat/acc-forum/avgnp/node31.html (1741 words)

	CS 612 - Graph Coloring (Site not responding. Last check: 2007-10-20)
		Besides vertex coloring there is edge coloring where you try to color the edges of a graph in such a way that no two edges with the same color share a vertex.
		The edge coloring of a graph is equivalent to the vertex coloring of its line graph.
		The line graph is the graph that is created by exchanging every edge by a vertex and connecting the vertices if the corresponding edges are connected.
	www.cosy.sbg.ac.at /~wdietl/study/cs612/pres.html (856 words)

	The Graph Coloring Problem (Site not responding. Last check: 2007-10-20)
		Coloring graphs using n colors is no easier than 3 (no surprise there).
		Color the adjacent vertices blue, then red, then blue, and so on throughout the graph.
		A collision of red and blue, as occurs in a pentagon, means the graph cannot be colored.
	www.mathreference.com /lan-cx-np,gcp.html (401 words)

	1.5.7 Vertex Coloring (Site not responding. Last check: 2007-10-20)
		Problem: Color the vertices of V with the minimum number of colors such that for each edge (i,j) \in E, vertices i and j have different colors.
		A coloring of the vertices of this graph assigns the variables to classes such that two variables with the same color do not clash and so can be assigned to the same register.
		The smallest number of colors sufficient to vertex color a graph is known as its chromatic number.
	www.cs.sunysb.edu /~algorith/files/vertex-coloring.shtml (313 words)

	Graph Coloring Example
		The graph (or vertex) coloring problem, which involves assigning colors to vertices in a graph such that adjacenct vertices have distinct colors, arises in a number of scientific and engineering applications such as scheduling, register allocation, optimization and parallel numerical computation.
		The problem is often to determine the minimum cardinality (the number of colors) of S for a given graph G or to ask whether it is able to color graph G with a certain number of colors.
		The algorithm uses a bucket sorter for the vertices in the graph where bucket is the degree.
	www.boost.org /libs/graph/doc/graph_coloring.html (637 words)

	Network Resources for Coloring a Graph
		Given an undirected graph, a clique of the graph is a set of mutually adjacent vertices.
		A minimum coloring of a graph is a coloring that uses as few different labels as possible.
		The minimum coloring problem is to assign a color to each item so that every incompatible pair is assigned different colors.
	mat.gsia.cmu.edu /COLOR/color.html (443 words)

	Computational Series: Graph Coloring and its Generalizations
		The bandwidth coloring problem: the graph coloring problem together with the constraint that the absolute value of the difference betwen r(i) and r(j) is at least d(i,j) for each edge.
		DSJR are geometric graphs, with DSJR..c being complements of geometric graphs.
		Given an n by n chessboard, a queen graph is a graph on n^2 nodes, each corresponding to a square of the board.
	mat.gsia.cmu.edu /COLORING02 (1747 words)

	Solution to a Graph Coloring problem (Site not responding. Last check: 2007-10-20)
		Problem : Provide a graph which does not contain any triangle and its chromatic number is 4.
		Before going to provide such a graph, I am going to show that : Two vertices can be made to be colored the same provided that the graph that contains these two vertices can be colored by 3 colors.
		It is obvious that if vertices A and B are colored with different colors then all vertices that are adjacent to both A and B must be colored with a different color, namely green.
	ce.sharif.edu /~rahban/Graph_Coloring.htm (187 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Definition: A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color.
		Definition: The chromatic number of a graph is the least number of colors needed for a coloring of this graph.
		Nonplanar graphs can have arbitrary large chromatic number Exmaple Theorem: A graph is a 2-colorable (i.e., bipartite) if and only if it has no cycles of odd lenght.
	www.cs.wm.edu /~nikos/cs243/Notes/L24.txt (780 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		A constraint satisfaction problem is a problem to find a consistent assignments of values to variables involved in the problem[5].
		One typical instance of CSP problem is graph coloring where the goal is to find a valid color assignment for every vertex in the graph.
		Two fundamental problems arose from this system are: 1) Finding an optimal ping-node configuration where a minimal set of actuators are selected to send detecting signals so long as the overall area can be covered.
	www.cs.wustl.edu /~xing/ColorCSP (2334 words)

	Amazon.ca: Books: Graph Coloring Problems (Site not responding. Last check: 2007-10-20)
		Every problem is stated in a self-contained, extremely accessible format, followed by comments on its history, related results and literature.
		This is typical of the problems cataloged in this book: a terse but formally correct statement of a problem followed by what is currently know, with full citations.
		This is an excellent reference for those who are interested in serious research in graph coloring.
	www.amazon.ca /exec/obidos/ASIN/0471028657 (689 words)

	Graph 4-coloring problem (Site not responding. Last check: 2007-10-20)
		We are given a graph G with the set V={1,...,n} of vertices and a set E of edges (unordered pairs of vertices).
		The objective is to find an assignment of colors to vertices so that for every edge, its vertices get different colors.
		We pick the ratio of the number of vertices to the number of edges so that about 50% of the random graphs we generated are sat.
	www.cs.uky.edu /~lliu1/experiment/4coloring/index.html (234 words)

	CP2002 Home page (Site not responding. Last check: 2007-10-20)
		The purpose of this Symposium is to encourage research on computational methods for combinatorial optimization problems, to evaluate alternative approaches using a common testbed, and to stimulate discussion on present and future directions in computational combinatorial optimization.
		This topic was chosen due to the wide applicability of graph coloring and the variety of solution approaches that have been proposed.
		In addition to the basic graph coloring problem, results are also solicited for the related problems of "multi-coloring" (assigning multiple colors to each node) and bandwidth allocation models (those with minimum difference requirements on the colors on adjacent nodes).
	www.cs.cornell.edu /cp2002 (1131 words)

	Graph Coloring Problems -- The archive. (Site not responding. Last check: 2007-10-20)
		Here are the archives for the book "Graph Coloring Problems" by Tommy R. Jensen and Bjarne Toft (Wiley Interscience 1995), dedicated to Paul Erdös.
		Graph Theory by Reinhard Diestel (Springer 1997) gives an introduction to general graph theory including chapters on coloring and integer flows.
		Digraphs: Theory, Algorithms and Applications by Jørgen Bang-Jensen and Gregory Gutin (Springer 2001) is a comprehensive text on directed graphs, containing material on the relations of graph orientations with coloring and integer flows, and with discussion of directed graph homomorphisms, among other topics.
	www.imada.sdu.dk /Research/Graphcol (170 words)

	GT 17/2 poz 1 (Site not responding. Last check: 2007-10-20)
		We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment.
		V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite these two invariants have nearly the same value for almost all graphs.
		O.V. Borodin, Problems of coloring and covering the vertex set of a graph by induced subgraphs.
	www.pz.zgora.pl /discuss/gt/17_2/g1.htm (3111 words)

	Application of the Graph Coloring Algorithm to the Frequency Assignment Problem (Site not responding. Last check: 2007-10-20)
		The problem is to assign channels to transmitters using the smallest span of frequency band while satisfying the requested communication quality.
		The k-coloring algorithm is modified to solve the frequency assignment problem.
		The performance of the proposed algorithm is tested with randomly generated graphs with different number of nodes, density types and graph types.
	heuristic.kaist.ac.kr /paper/Application%20of%20the%20Graph%20Coloring%20Algorithm%20to%20the%20Frequency%20Assignment%20Problem.htm (123 words)

	earthmoo (Site not responding. Last check: 2007-10-20)
		The lower bound is easily demonstrated by drawing 8 states, any one adjacent to any other on either the earth or the moon.
		The dual form of the problem asks for the maximum chromatic number among all graphs of thickness two (a thickness-two graph is one having an edge partition into two planar graphs, one on the earth and one on the moon).
		The problem generalizes to ask for the maximum chromatic number of all graph of thickness t.
	www.emba.uvm.edu /%7Earchdeac/problems/earthmoo.htm (399 words)

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