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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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Later, the Fritsch graph and Soifer graph provide two smaller counterexamples.<ref>Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the Four-Color Theorem?" Congr. Numer. 164, 159-175, 2003.</ref>
 
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&nbsp;4.
Image:Errera graph 6color edge.svg|The [[chromatic index]] of the Errera graph is&nbsp;6.
Image:Errera graph.svg|The Errera graph is [[Planar graph| planar]].
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:4-chromatic graphs]]

Latest revision as of 06:38, 13 April 2013

Template:Infobox graph

In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges discovered by Alfred Errera.[1] Published in 1921, it provides an example of how Kempe's proof of the four color theorem cannot work.[2][3]

Later, the Fritsch graph and Soifer graph provide two smaller counterexamples.[4]

The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5-vertex-connected graph and a 5-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 −(x2−2x−5)(x2+x−1)2(x3−4x2−9x+10)(x4+2x3−7x2−18x−9)2.

Gallery

References

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  2. ↑ Errera, A. "Du coloriage des cartes et de quelques questions d'analysis situs." Ph.D. thesis. 1921.
  3. ↑ Peter Heinig. Proof that the Errera Graph is a narrow Kempe-Impasse. 2007.
  4. ↑ Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the Four-Color Theorem?" Congr. Numer. 164, 159-175, 2003.
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