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Topic: Graph coloring

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	Coloring a Graph
		In graph coloring, the name of the game is to color the vertices using the fewest number of colors (the only restriction being that nodes joined by an edge cannot be colored with the same color).
		The purpose of the applet is to develop the user's skill in finding optimal colorings (colorings that use the minimal number of colors), or at least "good" colorings of the nodes of a graph.
		Coloring can be done either by picking a color from a checkbox at the bottom, or by cycling through the colors.
	www.cut-the-knot.org /Curriculum/Combinatorics/ColorGraph.shtml (291 words)

	PlanetMath: graph theory (Site not responding. Last check: )
		Graph theory is the branch of mathematics that concerns itself with graphs.
		Now, a (finite) graph is usually thought of as a subset of pairs of elements of a finite set (called vertices), or more generally as a family of arbitrary sets in the case of hypergraphs.
		In these type of colorings, one colors vertices (edges) of a graph so that no two vertices of the same color are adjacent (resp.
	planetmath.org /encyclopedia/GraphTheory.html (0 words)

	Graph coloring - Wikipedia, the free encyclopedia
		Graph coloring is not to be confused with graph labeling, which is an assignment of labels, usually also in the form of numbers, to vertices or edges.
		Graph coloring is still a very active field of research.
		The problem of coloring a graph has found a number of applications such as scheduling, register allocation in compilers, frequency assignment in mobile radios, and pattern matching.
	en.wikipedia.org /wiki/Graph_coloring (1568 words)

	Graph theory - Wikipedia, the free encyclopedia
		In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection.
		In 1852 Francis Guthrie posed the four color problem which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color.
		Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values.
	en.wikipedia.org /wiki/Graph_theory (1279 words)

	PlanetMath: colouring problem
		The colouring problem is to assign a colour to every vertex of a graph such that no two adjacent vertices have the same colour.
		Graph colouring problems have many applications in such situations as scheduling and matching problems.
		coloring problem, colour, color, graph colouring, graph coloring
	planetmath.org /encyclopedia/GraphColoring.html (0 words)

	Computational Series: Graph Coloring and its Generalizations
		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).
		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)

	WPI Computer Science - Efficient Graph Coloring Algorithms via Bottom-up Ordering & Subcubic Graph Structure
		A proper coloring of a graph is a partition of its elements in such a way that no two adjacent or incident elements belong to the same set in the partition.
		It is a vertex coloring, an edge coloring, or a total coloring, according as the elements to be partitioned are the vertices alone, the edges alone, or both the vertices and edges, respectively.
		A list coloring is a proper coloring subject to an extra condition that a color to be assigned to an element must come from that element's set ("list") of colors priorly associated with that element as part of the input.
	www.cs.wpi.edu /News/Colloquium/20023/20030131.html (0 words)

	Vertex Coloring (Site not responding. Last check: )
		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.
		As in the previous algorithm for planar graphs, vertices are colored sequentially, with the colors chosen in response to colors already assigned in the vertex's neighborhood.
		Color interchange is a win in terms of producing better colorings, at a cost of increased time and implementation complexity.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE178.HTM (0 words)

	Graph Theory Lesson 8
		What this does is to color the vertices of the graph using as few colors as possible and making sure that adjacent vertices always have different colors.
		The number of colors used is called the chromatic number of the graph.
		Graph coloring can be used to solve problems involving scheduling and assignments.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson8.htm (354 words)

	DIMACS/DIMATIA/Renyi Working Group on Graph Colorings and their Generalizations
		One of the central topics in graph theory is that of graph coloring.
		Applications of graph theory have led to fascinating generalizations of the notion of graph coloring, with motivation coming from problems of channel assignment in communications, traffic phasing, fleet maintenance, task assignment, and other practical problems.
		Here, we assign nonnegative integer channels to the vertices of a graph so that if two vertices are joined by an edge in the graph, their channels differ by at least two and if the two vertices have a common neighbor, then their channels differ.
	dimacs.rutgers.edu /Workshops/GraphColor/main.html (2333 words)

	Graph Coloring
		A graph is a collection of vertices (dots) and edges (lines) which connect the vertices.
		One portion of graph theory deals with coloring the vertices of a graph so that adjacent (joined) vertices are not colored the same.
		This graph is one of an infinite family of graphs which confirms a conjecture of G.A.Dirac from 1970.
	www.ma.iup.edu /cgi-bin/gallery?id=10 (0 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.kanadas.com /ccm/coloring (0 words)

	Soft Graph Coloring
		Soft graph coloring is a generalization of traditional graph coloring: the objective is to assign a color to each node in an undirected graph so that the number of edges that connect nodes of the same color is minimized.
		The essential objective of a soft graph colorer is to find an assignment of colors to nodes that minimizes the number of color conflicts, that is, the number of edges that connect nodes of the same color.
		In the conflicts diagram, for two colors, the theoretical lowest value that could be achieved for conflicts is 50%; the colorer achieves a value of around 58% after 50 steps and a mean of 65% (for moderate activation values); in the long term, it achieves a value of around 52%.
	ants.kestrel.edu /soft-graph-coloring/index.html (0 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 /~joe/Coloring (689 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.
		Constructing graph algorithms with BGL we have shown how to write this algorithm in the generic programming paradigm.
	www.boost.org /libs/graph/doc/graph_coloring.html (637 words)

	Graph Coloring
		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.uva.es /p/v1/193.html (224 words)

	The Artificial Unger Graph Coloring Applet
		The first graph is a planar bi-directional one, and thus the graph coloring problem becomes the map coloring problem.
		A complete graph is defined as one in which each vertex is connected to all the other vertices in the graph.
		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 (0 words)

	Joseph Culberson's Graph Coloring Resources Page (Site not responding. Last check: )
		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.
	web.cs.ualberta.ca /~joe/Coloring/index.html (689 words)

	[No title] (Site not responding. Last check: )
		In 1992, DIMACS included graph coloring as one of the three NP hard problems (the other two are maximum clique and satisfiability) for its second implementation challenge.
		This links list the graph coloring benchmarks for both random graphs and graphs derived from real-life problems, such as register allocation for variables in real codes, class scheduling graphs, and Latin square problem.
		While solutions for random graph, especially the one with 1,000 vertices and an edge probability slightly larger than 0.5 (known as the DIMACS challenge graph, instance DSJC1000.5) are still open.
	vlsicad.ucsd.edu /GSRC/bookshelf/Slots/GraphColoring (0 words)

	A Deterministic Solution to the Graph Coloring Problem
		Given an Undirected Graph G=(V, E) where V= Set of Vertices and E=Set of Edges, it is required to find out an assignment of colors to vertices, such that no two vertices which are connected by an edge would get the same color.
		But as we start assigning colors to the vertices, the number of available colors to the remaining vertices would also start reducing, depending on the existence of edges between vertices.
		And, whenever a color is assigned to a vertex, that color has to be marked with × for all the adjacent vertices, and this takes O(n) time where n = no. of the vertices.
	www.geocities.com /krishnapg/graphcoloring.html (0 words)

	biology - Graph coloring
		In general, techniques for graph coloring concentrate on finding the least number of colors needed to color the graph, called its chromatic number χ.
		For example the chromatic number of a complete graph of n vertices (a graph with an edge between every two vertices), is n.
		, for any planar graph G. This famous result, called the four-color theorem, was stated by P. Heawood in 1890, but remained unproven until 1976, when it was established by Kenneth Appel and Wolfgang Haken at the University of Illinois.
	www.biologydaily.com /biology/Graph_coloring (982 words)

	05C: Graph theory
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
		A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
		Determining the genus of a graph is NP-complete.
	www.math.niu.edu /~rusin/known-math/index/05CXX.html (0 words)

	New Page 1
		The colorings obtained by implementing the earlier version of this algorithm admitted a pattern which led to a partial proof of the conjecture.
		In the earlier version of the algorithm, the first member of a color class was simply a vertex that was not previously chosen having smallest label.
		Because of the relationship between colorations and independent sets, it is hoped that this function might aid in the study of colorations.
	nsm1.nsm.iup.edu /jjl/GraphColoring/index.htm (0 words)

	Andrei Lopatenko's Blog (Site not responding. Last check: )
		Given a graph (V,E) determine a minimal graph (V, E) such that there is a (directed) path from i to j in G iff there is a (directed) path from i to j in G.
		GRAPH ENCODABILITY Given directed graphs (V,A) and (V',A') is the an encoding of the first graph into the second one: a one to one function V->V', such tht for each edge \in A, there exists a dirfected path in A' This problem is solvable in PTIME for undirected graphs.
		One of the branches of the algorithmic graph theory is a set of problems alike what is the minimum nuber of probes of the adjacency matrix of the graph is required in the worst case, in order to determine whether the graph posseses a given property P. Of course, the only nontrivial properties are considered.
	andrei.lopatenko.com (5729 words)

	Graph Coloring Problems -- The archive.
		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)

	Graph Three Coloring
		Now the interesting thing about this graph is that the vertex on the left has to be a digit, the vertex on the right has to be a digit, and they have to be different.
		To do that we create a graph that can represent the current state, duplicate it as many times as we require steps for the computation, and then connect each graph to the next with the logic that converts this state into the next state.
		When you force the colorings on the input nodes and then color the graph, the output node will be colored 1 for Yes, this is a solution or 0 for No, this is not a solution.
	c2.com /cgi/wiki?GraphThreeColoring (0 words)

	An application of Iterated Local Search to Graph Coloring - Chiarandini, Stutzle (ResearchIndex)
		47 Genetic and hybrid algorithms for graph coloring (context) - Fleurent, Ferland - 1996
		39 Graph coloring with adaptive evolutionary algorithms - Eiben, van der Hauw et al.
		10 Order-based genetic algorithms and the graph coloring proble..
	citeseer.ist.psu.edu /chiarandini02application.html (634 words)

	Distributional Graph Edge Coloring
		Certain graph structures are needed for constructing the reduction.
		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)

	BackgroundMaterial (Site not responding. Last check: )
		So an interval graph is a special case of an intersection graph in which each vertex corresponds of a set and two vertices are adjacent if and only if their corresponding sets overlap.
		Let G_1 be the graph induced by the vertices in C plus x and y with the edge xy added if it is not already present in G. Also, let G_2 be the graph obrtained from G by deleting C and again making x and y adjacent.
		Suppose color a is missing at x and color b is missing at y.
	www.math.gatech.edu /~trotter/Section6-Coloring.htm (0 words)

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