en>Thore Husfeldt: added Category:4-chromatic graphs using HotCat

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added Category:4-chromatic graphs using HotCat

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			{{infobox graph
			\| name = Errera graph
			\| image = [[Image:Errera graph alt.svg\|220px]]
			\| image_caption = The Errera graph
			\| namesake = Alfred Errera
			\| vertices = 17
			\| edges = 45
			\| automorphisms = 20 ([[dihedral group\|D]]<sub>10</sub>)
			\| girth = 3
			\| radius = 3
			\| diameter = 4
			\| chromatic_number = 4
			\| chromatic_index = 6
			\| properties = [[Planar graph\|Planar]]<br>[[Hamiltonian graph\|Hamiltonian]]<ref>{{MathWorld\|urlname=HamiltonianGraph\|title=Hamiltonian Graph}}</ref>
			}}

			In the [[mathematics\|mathematical]] field of [[graph theory]], the '''Errera graph''' is a graph with 17 [[vertex (graph theory)\|vertices]] and 45 [[edge (graph theory)\|edge]]s discovered by Alfred Errera.<ref>{{MathWorld\|urlname=ErreraGraph\|title=Errera graph}}</ref> Published in 1921, it provides an example of how [[Alfred Kempe\|Kempe]]'s proof of the [[four color theorem]] cannot work.<ref>Errera, A. "Du coloriage des cartes et de quelques questions d'analysis situs." Ph.D. thesis. 1921.</ref><ref>Peter Heinig. [http://www-m9.ma.tum.de/foswiki/pub/Allgemeines/PeterHeinig/erreraGraphIsNarrowProof.pdf Proof that the Errera Graph is a narrow Kempe-Impasse]. 2007.</ref>

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			The Errera graph is [[Planar graph\| planar]] and has [[chromatic number]] 4, [[chromatic index]] 6, [[radius (graph theory)\|radius]] 3, [[diameter (graph theory)\|diameter]] 4 and [[girth (graph theory)\|girth]] 3. All its vertices are of degree 5 or 6 and it is a 5-[[k-vertex-connected graph\|vertex-connected graph]] and a 5-[[k-edge-connected graph\|edge-connected graph]].

			==Algebraic properties==
			The Errera graph is not a [[vertex-transitive graph]] and its full automorphism group is isomorphic to the [[dihedral group]] of order 20, the group of symmetries of a [[decagon]], including both rotations and reflections.

			The [[characteristic polynomial]] of the Errera graph is <math>-(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2</math>.

			==Gallery==
			<gallery>
			Image:Errera graph 4COL.svg\|The [[chromatic number]] of the Errera graph is 4.
			Image:Errera graph 6color edge.svg\|The [[chromatic index]] of the Errera graph is 6.
			Image:Errera graph.svg\|The Errera graph is [[Planar graph\| planar]].
			</gallery>

			== References ==
			{{reflist}}

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

en>Citation bot 1 at 18:09, 10 March 2011

2011-03-10T18:09:11Z

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Brinkmann graph - Revision history

en>Thore Husfeldt: added Category:4-chromatic graphs using HotCat

en>Citation bot 1 at 18:09, 10 March 2011