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Topic: Fractional coloring

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	Visualizing complex analytic functions using domain coloring
		For the sake of example, our first coloring will be very simple: let the point w=4 be red, the point w=−1 blue, the point w=0 green, and the rest of the w plane white (see figure 1, where the axes are shown for reference only; they don't belong to the coloring).
		Domain coloring perhaps doesn't do justice to all their properties, but let's look at a few things that can be seen.
		With our use of colors to keep track of arg f (z), this is the same as saying that the colors sweep through the color gradient N − P times as we move around γ.
	www.mai.liu.se /~halun/complex/domain_coloring-unicode.html (4509 words)

	[No title]
		The purpose of this note is to show that \kdcs are an instance of the more general concept of fractional nowhere-zero flows in regular matroids.
		It is well known that, in the setting of matroids, vertex colorings and nowhere-zero flows are dual concepts.
		The {\em fractional chromatic number} $\chi^f(G)$ is defined to be the least total weight of any weighted independent cover of $G$.
	www.math.sfu.ca /~goddyn/Papers/943-kd-colorings-fractional-flows.tex.txt (2236 words)

	The Geometry Junkyard: Coloring
		Fractional Graph Theory, a rational approach to the theory of graphs, Edward R. Scheinerman and Daniel Ullman, Johns Hopkins.
		Ivars Peterson reports on a new proof by Tom Sibley and Stan Wagon that the rhomb version of the tiling is 3-colorable; A proof of 3-colorability for kites and darts was recently published by Robert Babilon [Discrete Mathematics 235(1-3):137-143, May 2001].
		This is closely related to my page on line arrangement coloring, since every Penrose tiling is dual to a "multigrid", which is just an arrangement of lines in parallel families.
	www.ics.uci.edu /~eppstein/junkyard/color.html (362 words)

	Fractions
		Students practice fraction skills by doubling, tripling, and halving recipes of their choice.
		A clear explanation of adding like fractions is followed by three pages of practice.
		Color groups of pictures to learn about fractional parts.
	www.abcteach.com /directory/basics/math/fractions (483 words)

	[No title]
		Density plots of the cumulative sums of the fractional parts minus one-half of the expressions
		The left graphic shows the real part and the coloring is according to the imaginary part.
		The right graphic shows a contour plot of the argument over the complex plane and the coloring is cyclic.
	functions.wolfram.com /Constants/Pi/visualizations/5.html (167 words)

	Wiley::Fractional Graph Theory: A Rational Approach to the Theory of Graphs
		Professors Scheinerman and Ullman begin by developing a general fractional theory of hypergraphs and move on to provide in-depth coverage of fundamental and advanced topics, including fractional matching, fractional coloring, and fractional edge coloring; fractional arboricity via matroid methods; and fractional isomorphism.
		The final chapter is devoted to a variety of additional issues, such as fractional topological graph theory, fractional cycle double covers, fractional domination, fractional intersection number, and fractional aspects of partially ordered sets.
		Supplemented with many challenging exercises in each chapter as well as an abundance of references and bibliographic material, Fractional Graph Theory is a comprehensive reference for researchers and an excellent graduate-level text for students of graph theory and linear programming.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471178640.html (297 words)

	Interactivate: Activities
		Students color numbers in Pascal's Triangle by rolling a number and then clicking on all entries that are multiples of the number rolled, thereby practicing multiplication tables, investigating number patterns, and investigating fractal patterns.
		Graphically determine the value of two given fractions represented as points on a number line then graphically find a fraction whose value is inbetween the value of the 2 given fractions and determine its value.
		Students represents fractions by coloring in the appropriate portions of either a circle or a square, then order those fractions from least to greatest.
	www.shodor.org /interactivate/activities/index.html (3277 words)

	RR-4094 : Fractional Coloring of Bounded Degree Trees
		We also show some relationships between the integral and fractional problems, prove some polynomial instances of the coloring problem, and derive a $1 + 5/3e$ approximation for the \wdm problem in symmetric directed trees, where $e$ is the classical Neper constant, improving on previous results.
		Finally we present computational results suggesting that fractional coloring is a good oracle for a branch and bound strategy for coloring dipaths in symmetric directed trees and cycles.
		Notre recherche est motivée par le problème de coloration de chemins orientés où il faut colorier les chemins orientés d'un graphe orienté avec un nombre de couleurs minimum en respectant la contrainte que deux chemins traversant le même arc doivent avoir des couleurs différentes.
	www.inria.fr /rrrt/rr-4094.html (467 words)

	COMPUTER SCIENCE TECHNICAL REPORT ABSTRACTS (Site not responding. Last check: 2007-08-02)
		The required number of colors is called the chromatic index of the graph.
		To do this we first formulate edge coloring as an integer program and we define the fractional chromatic index to be the optimum of its linear programming relaxation.
		So it would be of interest to determine for which classes of simple graphs is the chromatic index equal to the ceiling of the fractional chromatic index as we can compute the chromatic index for graphs in these classes.
	reports-archive.adm.cs.cmu.edu /anon/1998/abstracts/98-176.html (259 words)

	Chapter 6 Nonwindow-based Applications (Site not responding. Last check: 2007-08-02)
		The input pdb file is converted to colored lines using the same routines as in xfit, except that the lines are colored by B-value.
		The output uses the standard GRINCH color scheme for interpreted edges (i.e., a main chain is green, a side chain is violet, etc.) except that unknown edges are colored in bins of density from blue to red (the same as the default color scheme for the first map in xfit).
		The output is colored in four bins of intensity from min to max: blue, cyan, yellow, and white.
	www.sdsc.edu /CCMS/Packages/XTALVIEW/XV6.doc.html (2652 words)

	Manpage (Site not responding. Last check: 2007-08-02)
		The output is colored in 4 bins of intensity from min to max: blue, cyan, yellow, white.
		To change the depth-cueing you change the four colors where red is the foreground and red4 is the color of the farthest point.
		Postscript coloring is always done in terms of RGB triples, and the color for the foreground is taken from colors.dat.
	www.sdsc.edu /CCMS/Packages/manpage.html (17543 words)

	ALCOMFT-TR-03-187 (Site not responding. Last check: 2007-08-02)
		Given a set of paths P on a graph G, the path coloring problem is to color the paths of P so that no two paths traversing the same edge of G are assigned the same color and the total number of colors used is minimized.
		Using optimal solutions to fractional path coloring, a natural relaxation of path coloring, on which we apply a randomized rounding technique combined with existing coloring algorithms, we obtain new upper bounds on the minimum number of colors sufficient to color any set of paths on any graph.
		The existential upper bounds are significantly better than existing ones provided that the cost of the optimal fractional path coloring is sufficiently large and the dilation of the set of paths is small.
	www.brics.dk /BRICS/ALCOM-FT/TR/ALCOMFT-TR-03-187.html (210 words)

	Research interests: Daniel Ullman (Site not responding. Last check: 2007-08-02)
		My special areas of interest within graph theory include the study of fractional analogues of integer-valued graph invariants, the interaction between combinatorial game theory and graphs, the representation of graphs by geometric objects, the chromatic theory of graphs (i.e., graph coloring), and the computational complexity of various graph problems.
		Fractional isomorphism of graphs, J. Graph Theory 132 (1994) 247-265 (with M. Ramana, E. Scheinerman).
		The fractional chromatic number of Mycielski's graphs, J. Graph Theory 19 (1995) 411-416 (with J. Propp, M. Larsen).
	www.gwu.edu /~math/research/ullman.html (182 words)

	Publications by William Trotter
		On-line coloring and recursive graph theory, SIAM J. Discrete Math.
		Colorings of diagrams of interval orders and alpha-sequences of sets, Discrete Math.
		Competitive colorings of oriented graphs, Electronic Journal of Combinatorics 8 (2001) no.2, Research Paper 12 (with H. Kierstead).
	www.math.gatech.edu /~trotter/E-Pubs.html (456 words)

	MINIMUM GRAPH COLORING
		A coloring of G, i.e., a partition of V into disjoint sets
		Approximable with an absolute error guarantee of 1 on planar graphs by the Four Color Theorem.
		The complementary maximization problem, where the number of ``not needed colors'', i.e.
	www.nada.kth.se /~viggo/wwwcompendium/node15.html (172 words)

	Fractional coloring - Wikipedia, the free encyclopedia
		A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets.
		Scheinerman, Edward R.; Ullman, Daniel H. Fractional graph theory.
		This page was last modified 18:35, 6 July 2006.
	en.wikipedia.org /wiki/Fractional_coloring (143 words)

	Papers Report (Site not responding. Last check: 2007-08-02)
		We study the problem of scheduling independent multiprocessor tasks, where for each task in addition to the processing time(s) there is a prespecified dedicated subset (or a family of alternative subsets) of processors which are required to process the task simultaneously.
		All these results are based on a nice relation between preemptive scheduling and fractional coloring of graphs.
		In contrast to the positive results, we also prove that the problems of computing optimal preemptive schedules for three-processor tasks or for bi-processor tasks with (possible several) alternative modes are strongly NP-hard.
	www.mfcs.sk /mfcs2000/abstracts/AccAbs1.html (189 words)

	[No title]
		The goal is to minimize the number of colors used by the coloring $\chi$.
		The main contribution is that the algorithm solves the general mixed fractional packing and covering problem (in contrast to pure fractional packing and pure fractional covering problems) and runs in time independent of the so-called width of the problem.
		We first present a general $O(\log m)$-deterministic algorithm for generating a fractional solution that satisfies the online connectivity or cut demands, where $m$ is the number of edges in the network.
	www.informatik.uni-kiel.de /inf/Jansen/conference/2004/WASC/abstracts.html (3038 words)

	A Column Generation Approach (SMEALSearch) - Pal,Rangaswamy,Giles,Debnath
		For Graph Coloring Anuj Mehrotra Department of Management Science School of...
		We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph).
		1 The fractional chromatic number of a graph and a constructio..
	gunther.smeal.psu.edu /5868.html (582 words)

	"Erdos-Faber-Lov\'asz Conjecture"
		A proper n-edge-coloring of a hypergraph assigns colors from an n-set to the edges so that edges sharing a vertex have distinct colors.
		Kahn and Seymour proved a fractional version of Conjecture 1, showing that the fractional chromatic number of such a graph is n.
		Chang W.I. and Lawler E. Edge coloring of hypergraphs and a conjecture of Erdos, Faber, and Lovasz.
	www.math.uiuc.edu /~west/openp/erdfablov.html (324 words)

	Cumberland Conference Abstracts
		Finding a legal coloring of the vertices of a graph using the minimum number of colors is one of the most computationally difficult problems in graph theory.
		After examining the effectiveness of pure genetic algorithms for graph coloring using both order based and coloring based encodings, we also hybridize the genetic algorithm with other heuristic techniques that have shown to be effective in attacking this problem.
		Many graph theoretic subset problems, such as, fractional domination, fractional independence, and fractional packing, can be formulated as Linear Programming (LP) problems, where the constraint matrix for the LP-problem is the closed neighborhood, adjacency, incidence,or total matrix.
	www.etsu.edu /MATH/abs1.htm (4330 words)

	Slope Tutorial Page 5
		The slope formula was used for all layers.
		The color gradient used for the bottom layer (Layer 1) is the same gray-scale gradient presented earlier.
		The next 3 images show the merge mode and the effects of adding a layer at a time, starting with layer 1.
	www.hiddendimension.com /Slope_Tutorial_5.html (69 words)

	[No title]
		Jansen, Approximation algorithms for fractional covering and packing problems, and applications, (invited talk), to appear in: FCT 2001, Riga.
		Gargano, A. Rescigno, Coloring Circular Arcs with Applications to WDM Routing, Workshop on Approximation and Randomization Algorithms in Communication Networks (ARACNE'00), July 2000, Geneva, Switzerland.
		T. Erlebach and K. Jansen, Conversion of coloring algorithms into maximum weight independent set algorithms, Workshop on Approximation and Randomization Algorithms in Communication Networks, ARACNE 2000, Geneva, Carleton Scientific, 2000, 135-146.
	www.dia.unisa.it /ARACNE/Pubbl.html (1222 words)

	AMCA: Polynomial-time Approximation Algorithms for Preemptive Resource Constrained Scheduling and Fractional Graph ... (Site not responding. Last check: 2007-08-02)
		We study resource constrained scheduling problems where the objective is to compute feasible preemptive schedules minimizing the makespan and using no more resources than what are available.
		Finally we present applications of the above results in fractional graph coloring and multiprocessor task scheduling.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/v/86.htm (160 words)

	meisme768's Xanga Site
		Coloring a graph = OK. Chromatic number of a graph is still fine.
		Then shake it around until all the dots are covered with a certain amount.
		So instead, I'll move on to the subject where coffee cups and donuts are the same, but rings aren't donuts.
	www.xanga.com /meisme768 (1443 words)

	Smooth Escape Iteration Counts
		In general, the idea is that for a fairly general sequence of orbits of an iterated function, a regulated and well-behaved series can be defined.
		In turn, the fractional iteration count can be used to provide a smooth coloring for Mandelbrot-based artwork.
		The claim made here is also that the finite part that remains after the removal of the divergent parts is a candidate for the smooth, fractional iteration escape count of an iterated function.
	linas.org /art-gallery/escape/math.html (745 words)

	LEARNING BASIC FRACTIONS
		The Mars Fraction Hunt is a fun game which requires you to find fractions of words to solve the puzzle.
		Complete the exercises on the first line which is "Identifying fractions with lines." Read the instructions and complete the practice exercise on identifying numerator and denominator.
		Basic Fractions is a quiz you will take to test your knowledge of fractions.
	www.rblewis.net /technology/EDU506/WebQuests/basicfractions/fractions.html (518 words)

	Technical Reports produced within CRESCCO
		Approximate path coloring with application to wavelength assignment in WDM optical networks.
		Fractional coloring of symmetric paths on binary trees.
		Approximate strong separation with application in fractional graph coloring and preemptive scheduling.
	www.ceid.upatras.gr /crescco/reports_1.htm (2899 words)

	Home (Site not responding. Last check: 2007-08-02)
		Graphically determine the value of 2 given fractions represented as points on a number line then graphically find a fraction whose value is in between the value of the 2 given fractions and determine its value.
		Students can view histograms for either built-in or user-specified data, and experiment with how the size of the class intervals influences the perceptions.
		Students choose one of three doors to experimentally determine the odds of winning the grand prize behind one of the doors, as in the TV program "Let's Make a Deal." Parameters: Staying or switching between the two remaining doors.
	chesterfield.k12.va.us /~mambrose/home/mathpractice.htm (2034 words)

	05C15 - Coloring of Graphs and Hypergraphs
		The Four Color Theorem (by MacTutor History of Math, Scotland)
		Fractional Graph Theory by Scheinerman.ER and Ullman.D John Wiley and Sons 1997
		On certain homomorphism-properties of graphs with applications to the conjecture of Hadwiger.
	www.cs.columbia.edu /~sanders/graphtheory/research/05C15.html (195 words)

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