en>Dthomsen8: /* Application */clean up, typo(s) fixed: , → , (2) using AWB

2014-01-23T01:46:02Z

Application: clean up, typo(s) fixed: , → , (2) using AWB

← Older revision		Revision as of 03:46, 23 January 2014
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	~~Oscar is what my~~ [~~http~~:~~//Goaskalice~~.~~Columbia.edu~~/~~what~~-do-~~genital~~-~~warts~~-~~look wife loves~~] to ~~contact me and I completely dig~~ that ~~name~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota~~ and ~~I love each over~~ the ~~counter~~ [~~http:~~/~~/wixothek~~.~~com/user/MBuckmast std home test~~] ~~test day residing here~~. ~~I utilized to be unemployed but now I am~~ a ~~librarian~~ and the ~~salary has been really fulfilling~~. ~~One~~ of the ~~extremely very best things in home std test kit~~ the ~~world for std testing~~ [~~http:~~/~~/nuvem~~.~~tk/altergalactica/AliceedMaurermy at home std testing~~] ~~home me~~ is ~~to do aerobics and I've been performing it for quite a while.~~<br><br>~~Also visit my web page:~~ [~~http://www~~.~~revleft~~.~~com~~/~~vb/member~~.~~php?u~~=~~160656 http~~:~~//www~~.~~revleft~~.~~com/~~]		{{infobox graph
			\| name = Grötzsch graph
			\| image = [[Image:Groetzsch-graph.svg\|180px]]
			\| namesake = [[Herbert Grötzsch]]
			\| vertices = 11
			\| edges = 20
			\| diameter = 2
			\| radius = 2
			\| girth = 4
			\| chromatic_number = 4
			\| chromatic_index = 5
			\| crossing_number = 5
			\| automorphisms = 10 ([[dihedral group\|D]]<sub>5</sub>)
			\| properties = [[Hamiltonian graph\|Hamiltonian]]<br>[[triangle-free graph\|Triangle-free]]
			}}
			In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Grötzsch graph''' is a [[triangle-free graph]] with 11 vertices, 20 edges, [[chromatic number]] 4, and [[Crossing number (graph theory)\|crossing number]] 5. It is named after German mathematician [[Herbert Grötzsch]], and its existence demonstrates that the assumption of planarity is necessary in [[Grötzsch's theorem]] (Grötzsch 1959) that every triangle-free [[planar graph]] is 3-colorable. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the [[Mycielskian]] of the previous graph in the sequence, starting from the [[null graph]]; this sequence of graphs was used by {{harvtxt\|Mycielski\|1955}} to show that there exist triangle-free graphs with arbitrarily large chromatic number. Therefore, the Grötzsch graph is sometimes also called the Mycielski graph or the Mycielski–Grötzsch graph. Unlike later graphs in this sequence, the Grötzsch graph is the smallest triangle-free graph with its chromatic number {{harv\|Chvátal\|1974}}.

			{{harvtxt\|Häggkvist\|1981}} used a modified version of the Grötzsch graph to disprove a conjecture of {{harvs\|txt\|first1=Paul\|last1=Erdős\|author1-link=Paul Erdős\|first2=Miklos\|last2=Simonovits\|year=1973}} on the chromatic number of triangle-free graphs with high degree. Häggkvist's modification consists of replacing each of the five degree-four vertices of the Grötzsch graph by a set of three vertices, replacing each of the five degree-three vertices of the Grötzsch graph by a set of two vertices, and replacing the remaining degree-five vertex of the Grötzsch graph by a set of four vertices. Two vertices in this expanded graph are connected by an edge if they correspond to vertices connected by an edge in the Grötzsch graph. The result of Häggkvist's construction is a 10-[[regular graph\|regular]] triangle-free graph with 29 vertices and chromatic number 4, disproving the conjecture that there is no 4-chromatic triangle-free ''n''-vertex graph in which each vertex has more than ''n''/3 neighbours.

			The Grötzsch graph has [[chromatic index]] 5, radius 2, girth 4 and diameter 2. It is also a 3-[[k-vertex-connected graph\|vertex-connected]] and 3-[[k-edge-connected graph\|edge-connected]] graph. The [[independence number]] is 5, and the [[domination number]] is 3.

			==Algebraic properties==
			The full automorphism group of the Grötzsch graph is [[Group isomorphism\|isomorphic]] to the [[dihedral group]] D<sub>5</sub> of order 10, the group of symmetries of a [[regular pentagon]], including both rotations and reflections.

			The [[characteristic polynomial]] of the Grötzsch graph is
			:<math>-(x-1)^5 (x^2-x-10) (x^2+3 x+1)^2.\ </math>

			==See also==
			*[[Chvátal graph]], another small triangle-free 4-chromatic graph.

			==References==
			*{{citation
			\| last = Chvátal \| first = Vašek
			\| authorlink = Václav Chvátal
			\| contribution = The minimality of the Mycielski graph
			\| title = Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973)
			\| date = 1974
			\| pages = 243–246
			\| publisher = Lecture Notes in Mathematics, Vol. 406, Springer-Verlag
			\| location = Berlin
			\| id = {{MathSciNet \| id = 0360330}}}}.
			*{{citation
			\| author1-link = Paul Erdős \| last1 = Erdős \| first1 = P. \| last2 = Simonovits \| first2 = M.
			\| title = On a valence problem in extremal graph theory
			\| journal = [[Discrete Mathematics (journal)\|Discrete Mathematics]]
			\| volume = 5
			\| year = 1973
			\| pages = 323–334
			\| id = {{MathSciNet \| id = 0342429}}
			\| doi = 10.1016/0012-365X(73)90126-X
			\| issue = 4}}.
			*{{citation
			\| last = Grötzsch \| first = Herbert
			\| title = Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel
			\| journal = Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe
			\| volume = 8
			\| year = 1959
			\| pages = 109–120
			\| id = {{MathSciNet \| id = 0116320}}}}.
			*{{citation
			\| last = Häggkvist \| first = R.
			\| title = Odd cycles of specified length in nonbipartite graphs
			\| booktitle = Graph Theory (Cambridge, 1981)
			\| date = 1981
			\| pages = 89–99
			\| id = {{MathSciNet \| id = 0671908}}}}.
			*{{citation
			\| last = Mycielski \| first = Jan
			\| authorlink = Jan Mycielski
			\| title = Sur le coloriage des graphs
			\| journal = Colloq. Math.
			\| volume = 3
			\| year = 1955
			\| pages = 161–162
			\| id = {{MathSciNet \| id = 0069494}}}}.

			== External links ==

			*{{mathworld \| title = Grötzsch Graph \| urlname = GroetzschGraph}}

			{{DEFAULTSORT:Grotzsch graph}}

			[[Category:Individual graphs]]
			[[Category:4-chromatic graphs]]

220.68.68.46: /* Example: pendulum */

2012-08-03T04:56:55Z

Example: pendulum

New page

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en>Dthomsen8: /* Application */clean up, typo(s) fixed: , → , (2) using AWB

220.68.68.46: /* Example: pendulum */