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Topic: Chromatic number

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	chromatic number
		In graph theory, the minimum number of colors needed to color (the vertices of) a connected graph so that no two adjacent vertices are colored the same.
		The chromatic number of a square, tube, or sphere, for example, is 4; in other words, it is impossible to place more than four differently-colored regions on one of these figures so that any pair has a common boundary.
		The chromatic number, in both senses just described, is 7 for the torus, 6 for the Möbius band, and 2 for the Klein bottle.
	www.daviddarling.info /encyclopedia/C/chromatic_number.html (241 words)

	PlanetMath: chromatic number
		The chromatic number of a graph is the minimum number of colours required to colour it.
		In general, however, finding the chromatic number of a large graph (and, similarly, an optimal colouring) is a very difficult (NP-hard) problem.
		This is version 5 of chromatic number, born on 2002-02-03, modified 2004-04-13.
	planetmath.org /encyclopedia/ChromaticNumber.html (143 words)

	PlanetMath: chromatic number of a metric space
		The chromatic number of a metric space is the minimum number of colors required to color the space in such a way that no two points at distance
		Alternatively, the chromatic number of a metric space is the chromatic number of a graph whose
		This is version 38 of chromatic number of a metric space, born on 2004-01-12, modified 2007-01-08.
	planetmath.org /encyclopedia/ChromaticNumberOfASpace.html (374 words)

	Chromatic number@Everything2.com
		In combinatorics, the chromatic number (gamma of G) of a graph is the smallest number of colors such that colors can be assigned to all nodes of the graph without having connected nodes with the same color.
		We can get another bound on the chromatic number of a graph G using the maximal degree (or valency) of the vertices in G. Suppose G is a graph in which the maximum degree of any vertex is d.
		The chromatic number of a graph is the smallest natural number that is not a root of the chromatic polynomial.
	everything2.com /index.pl?node_id=891966 (1034 words)

	Chromatic numbers explained_page 1_@TheCipher.com
		0° — 12° for one octave of chromatic scale-tones.
		That is, the chromatic scale having a C tonic or root.
		That chromatically numbered letter-line can be used to determine the true intervalic distance and true numeric values, in half-steps or semitones, of any lettered tone(s) of a C tonic or root.
	www.thecipher.com /chromatic-numbers_1.html (1832 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.
		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.
		For example the chromatic number of a complete graph of n vertices (a graph with an edge between every two vertices), is n.
	en.wikipedia.org /wiki/Graph_coloring (1104 words)

	Colorful Mathematics: Part IV
		(Chromatic numbers were defined in an earlier column.) Obviously, it is possible to color the vertices of the graph above with 8 colors because one can color each vertex with a different color.
		In general, the chromatic number of a graph is not necessarily a factor of the number of vertices of the graph.
		Even when the chromatic number does divide the number of vertices of the graph being colored, one may not be able to achieve a coloring where each color is used equally often.
	www.ams.org /featurecolumn/archive/colorapp2.html (774 words)

	Graph Theory Lesson 8
		The number of colors used is called the chromatic number of the graph.
		We use X(G) to denote the chromatic number of the graph G.
		Use the program to find the chromatic number of each of the five platonic graphs.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson8.htm (354 words)

	chromatic number :: Mathematical Dictionary, VeryPrime.com
		The chromatic number is the smallest number n such that graph G has an n-coloring.
		The graph with the chromatic number n is called n-chromatic graph.
		The chromatic number of the suspension of G from complete graph `K_1` is chr(G)+1.
	www.veryprime.com /dict/chromatic_number.php (88 words)

	Discrete Math/Theory of Computing seminar May 6, 2003 (Site not responding. Last check: )
		The problem CNP of finding chromatic number of the plane has been credited in print to P. Erdos, M. Gardner, H. Hadwiger, L. Moser and E. Nelson.
		In a joint work with Saharon Shelah the presenter points out circumstances under which the chromatic number of the plane would depend upon a system of axioms chosen for set theory.
		In the same joint work a graph on the real line is constructed, whose chromatic number is affected dramatically by a system of axioms chosen for sets.
	www.math.rutgers.edu /~saks/DMSEM/S03/0506.html (183 words)

	Colorful Mathematics/Teacher
		As with the chromatic number, the goal of this game is to color the vertices of a graph with the smallest possible number of colors, such that two vertices connected by an edge receive different colors.
		This number is always at least as big as the chromatic number of the graph, but often much larger.
		This number is at least as large as the chromatic number of the graph, and usually much bigger.
	www.math.ucalgary.ca /~laf/colorful/teacher.html (2018 words)

	Graph Theory Lesson 8
		The number of colors used is called the chromatic number of the graph.
		We use X(G) to denote the chromatic number of the graph G.
		Use the program to find the chromatic number of each of the five platonic graphs.
	www.utc.edu /~cpmawata/petersen/lesson8.htm (354 words)

	On Graphs with Bounded Chromatic Number
		A family $\cal C$ of graphs is said to be a family of graphs with bounded chromatic number if there is a function $f$ such that for every graph $G$ of $\cal C$, we have $\chi (H) \leq f (\omega (H))$ for any induced subgraph $H$ of $G$.
		Here $\chi (G)$ denotes the chromatic number of $G$ and $\omega (G)$ denotes the number of vertices in a largest clique of $G$.
		The class of perfect graphs is a well known class of graphs with bounded chromatic number.
	dmawww.epfl.ch /roso.mosaic/ismp97/ismp_abs_1324.html (110 words)

	[No title]
		Here is a pretty good list of references for the chromatic number of the plane (i.e., how many colors do you need so that no two points 1 away are the same color) up to around 1982 (though the publication dates are up to 1985).
		This asks for the chromatic number of the graph where two points in R^2 are connected if they are distance 1 apart.
		Let this chromatic number be chi(2) and in general let chi(n) be the chromatic number of R^n.
	www.ics.uci.edu /~eppstein/junkyard/plane-color.html (1140 words)

	Graph Theory Open Problems
		This number is also called ``the chromatic number of the plane.''
		An answer to this question may help to solve the "chromatic number of the plane" problem.
		It is known that this is not true if you remove the "bipartite" condition, but the smallest known such graph which is not Hamiltonian has 38 vertices, as shown to the right.
	dimacs.rutgers.edu /~hochberg/undopen/graphtheory/graphtheory.html (705 words)

	Chromatic Number Algorithms
		The chromatic number of a graph is the minimum number of color necessary to color a graph.
		A coloration of the graph and the chromatic number of the graph.
		Number of color used to color the neighbours of u.
	www.infres.enst.fr /~csernel/Chromatic.htm (257 words)

	Chromatic numbers explained_page 2_@TheCipher.com
		The Cipher System’s numbers and number-formula are easy to distinguish from music’s standard number-formula; The Cipher System’s formula numbers always have degree symbols (°) superscript to the right of each formula digit, and they never have flats or sharps.
		In the diatonic realm (a seven-thing oriented and one-based environment) the number seven is the octave multiplier.
		In the chromatic environment (zero-based and twelve-thing oriented) the number twelve is the octave multiplier.
	www.thecipher.com /chromatic-numbers_2.html (2643 words)

	On the Delta(d)-chromatic number of a complete balanced multipartite graph
		In this paper we solve (approximately) the problem of finding the minimum number of colours with which the vertices of a complete, balanced, multipartite graph G may be coloured such that the maximum degrees of all colour class induced subgraphs are at most some specified natural number d.
		The minimum number of colours in such a colouring is referred to as the Delta(d)-chromatic number of G.
		The problem of finding the Delta(d)-chromatic number of a complete, balanced, multipartite graph has its roots in an open graph theoretic characterisation problem and has applications conforming to the generic scenario where users of a system are in conflict if they require access to one or more of the same resources.
	dip.sun.ac.za /~vuuren/abstracts/abstr_deltachromatic.htm (334 words)

	Game Chromatic Number of Graphs - Dinski, Zhu (ResearchIndex) (Site not responding. Last check: )
		By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs.
		In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the game chromatic number of a planar graph is at most 30, which improves the previous known upper bound for the game...
		1 Acyclic chromatic numbers of graphs (context) - Kostochka, Sopena et al.
	citeseer.ist.psu.edu /dinski98game.html (523 words)

	onechrom
		The 1-chromatic number of the surface is the maximum chromatic number over all graphs 1-embeddable in that surface.
		Borodin showed that the 1-chromatic number of the sphere is 6; for the best proof see [B].
		Korzhik believes he has a long proof that the 1-chromatic number of a nonorientable surface and the Heawood-type bound always differ by at most one, but the argument may be too involved for publication without settling the problem in its entirety.
	www.emba.uvm.edu /~archdeac/problems/onechrom.htm (383 words)

	Circular Chromatic Number of Planar Graphs of Large Odd Girth (ResearchIndex) (Site not responding. Last check: )
		The restriction of this conjecture to planar graphs is equivalent to the statement that planar graphs of girth at least 4k have circular chromatic number at most 2 + 1 k.
		10 The circular chromatic number of series-parallel graphs of l..
		10 Circular chromatic number of series-parallel graphs of large..
	citeseer.ist.psu.edu /zhu01circular.html (516 words)

	chrompln (Site not responding. Last check: )
		What is the minimum number of colors needed to color the plane so that no two points of the same color are exact one unit apart?
		A widget that shows that the chromatic number of the plane is at least 4.
		Thus, we know that the chromatic number of the plane is either 4, 5, 6, or 7 at this point.
	www.mathpuzzle.com /chrompln.html (212 words)

	Amazon.com: "edge chromatic number": Key Phrase page
		Berge includes a treatment of the fractional matching number and the fractional edge chromatic number.' Two decades have seen a great deal of development in the field of fractional graph theory and the time is...
		Thus the chromatic polynomial and vertex coloring functions can be applied to edge colorings.
		The edge chromatic number of a graph must be at least A, the largest degree vertex of the graph,...
	www.amazon.com /phrase/edge-chromatic-number (300 words)

	Sixth Czech-Slovak International Symposium 2006
		Andrew Goodall: The discrete Fourier transform and the number of nowhere-zero 4-flows of a graph
		Leonid L. Ivanov: An estimate on the chromatic number of the chromatic number of the 4-space.
		Nicolas Roussel: circular chromatic number of sparse graphs
	kam.mff.cuni.cz /~cs06/accepted.html (1242 words)

	chromatic number - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "chromatic number" is defined.
		Chromatic Number : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include chromatic number: chromatic number and girth, chromatic number of a space, edge chromatic number, proof of chromatic number and girth
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=chromatic+number (116 words)

	On the Oriented Game Chromatic Number (ResearchIndex) (Site not responding. Last check: )
		We prove that every oriented path has oriented game chromatic number at most 7 (and this bound is tight), that every oriented tree has oriented game chromatic number at most 19 and that there exists a constant t such that every oriented outerplanar graph has oriented game chromatic number at most t.
		49 the chromatic number of oriented graphs - Sopena - 1997 ACM
		6 the game chromatic number of some classes of graphs (context) - Faigle, Kern et al.
	citeseer.ist.psu.edu /nesetril00oriented.html (449 words)

	Colorful Mathematics/Games/Chromatic
		This game deals with the general concept of a graph and its chromatic number.
		If you draw a bunch of circles, which we call vertices, and join some of them by lines, which we call edges, you get what is called a graph.
		If you now color the vertices in such a way that those joined by an edge receive different colors, then the smallest number of colors that can be used is called the chromatic number of the graph.
	math.ucalgary.ca /~laf/colorful/chromatc.html (400 words)

	Amazon.com: "chromatic number": Key Phrase page
		things, that for all natural numbers g >_ 3 and k >_ 3 there exist graphs with girth g and chromatic number k.
		context, lie defined the class of perfect graphs, and noted that for certain channels, the Shannon capacity was simply the chromatic number of the associated perfect graph.
		such that each vertex gets a unique color and no two adjacent vertices get the same color is called the chromatic number (or vertex chromatic number) of the graph.
	www.amazon.com /phrase/chromatic-number (309 words)

	[No title]
		Plainly the number of colors required grows without bound.
		Saaty and Kainen's book on the 4CT gives the number of colors required for every surface.
		They are the same as the maximum number of vertices in a complete graph that can be embedded in the surface (although this is far from obvious).
	www.math.niu.edu /~rusin/known-math/99/chromatic (336 words)

	On the Complexity of Finding the Chromatic Number of a Recursive Graph I: The Bounded Case - Beigel, Gasarch ...
		In this section, we consider the problem of computing the chromatic number, G) of a graph when we are not given an a priori bound on...
		On the complexity of finding the chromatic number of a recursive graph II: The unbounded case.
		25 the complexity of finding the chromatic number of a recursiv..
	citeseer.ist.psu.edu /269707.html (784 words)

	Towards Optimal Lower Bounds For Clique and Chromatic Number - Engebretsen, Holmerin (ResearchIndex) (Site not responding. Last check: )
		In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that unless NP), neither Max Clique nor Min Chromatic Number can be approximated within n.
		Engebretsen, J. Holmerin, Towards optimal lower bounds for clique and chromatic number, Theoretical Computer Science, to appear.
		62 the hardness of approximating the chromatic number - Khanna, Linial et al.
	citeseer.lcs.mit.edu /546271.html (528 words)

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