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| name = Wagner graph
| image = [[File:Wagner graph ham.svg|220px]]
| image_caption = The Wagner graph
| namesake = [[Klaus Wagner]]
| vertices = 8
| edges = 12
| automorphisms = 16 ([[dihedral group|D]]<sub>8</sub>)
| girth = 4
| diameter = 2
| radius = 2
| chromatic_number = 3
| chromatic_index = 3
| notation = ''M''<sub>''8''</sub>
| properties = [[Cubic graph|Cubic]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Triangle-free graph|Triangle-free]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[Toroidal graph|Toroidal]]<br>[[Apex graph|Apex]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Wagner graph''' is a 3-[[regular graph]] with 8 vertices and 12 edges.<ref>{{cite book
| author = [[John Adrian Bondy|Bondy, J. A.]]; [[U. S. R. Murty|Murty, U. S. R.]]
| title = Graph Theory
| year = 2007
| publisher = Springer
| isbn = 978-1-84628-969-9
| pages = 275–276}}</ref> It is the 8-vertex [[Möbius ladder]] graph.


==Properties==
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As a Möbius ladder, the Wagner graph is [[Planar graph|nonplanar]] but has [[Crossing number (graph theory)|crossing number]] one, making it an [[apex graph]]. It can be embedded without crossings on a [[torus]] or [[projective plane]], so it is also a [[toroidal graph]]. It has girth 4, diameter 2, radius 2, [[chromatic number]] 3, [[chromatic index]] 3 and is both  3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]].
 
The Wagner graph has 392 [[spanning tree]]s; it and the [[complete graph]] ''K''<sub>3,3</sub> have the most spanning trees among all cubic graphs with the same number of vertices.<ref>{{cite arxiv
  | author = Jakobson, Dmitry; Rivin, Igor
  | title = On some extremal problems in graph theory
  | year = 1999
  | eprint = math.CO/9907050
  | postscript = .}}</ref>
 
The Wagner graph is a [[vertex-transitive graph]] but is not [[edge-transitive graph|edge-transitive]]. Its full automorphism group is isomorphic to the [[dihedral group]] D<sub>8</sub> of order 16, the group of symmetries of an [[octagon]], including both rotations and reflections.
 
The [[characteristic polynomial]] of the Wagner graph is <math>(x-3)(x-1)^2(x+1)(x^2+2x-1)^2</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
 
The Wagner graph is [[triangle-free graph|triangle-free]] and has [[independence number]] three, providing one half of the proof that the [[Ramsey number]] ''R''(3,4) (the least number ''n'' such that any ''n''-vertex graph contains either a triangle or a four-vertex independent set) is&nbsp;9.<ref>{{Cite book|first=Alexander|last=Soifer|year=2008|title=The Mathematical Coloring Book|publisher=Springer-Verlag|isbn=978-0-387-74640-1|page=245|postscript=.}}.</ref>
 
==Graph minors==
Möbius ladders play an important role in the theory of [[graph minor]]s. The earliest result of this type is a 1937 theorem of [[Klaus Wagner]] (part of a cluster of results known as [[Wagner's theorem]]) that graphs with no ''K''<sub>5</sub> minor can be formed by using [[clique-sum]] operations to combine planar graphs and the Möbius ladder ''M''<sub>8</sub>.<ref>{{cite journal
  | author = Wagner, K. | authorlink = Klaus Wagner
  | title = Über eine Eigenschaft der ebenen Komplexe
  | journal = Mathematische Annalen
  | volume = 114
  | issue = 1
  | year = 1937
  | pages = 570–590
  | doi = 10.1007/BF01594196
  | postscript = .}}</ref> For this reason ''M''<sub>8</sub> is called the Wagner graph.
 
The Wagner graph is also one of four minimal [[forbidden minor]]s for the graphs of [[treewidth]] at most three (the other three being the [[complete graph]] ''K''<sub>5</sub>, the graph of the [[regular octahedron]], and the graph of the [[pentagonal prism]]) and one of four minimal forbidden minors for the graphs of [[branchwidth]] at most three (the other three being ''K''<sub>5</sub>, the graph of the octahedron, and the [[hypercube graph|cube graph]].<ref>{{Cite journal
  | last = Bodlaender | first = Hans L. | authorlink = Hans L. Bodlaender
| title = A partial ''k''-arboretum of graphs with bounded treewidth
| journal = Theoretical Computer Science
| volume = 209 | issue = 1–2 | pages = 1–45 | year = 1998 | doi = 10.1016/S0304-3975(97)00228-4
| postscript = <!--None-->}}.</ref><ref>{{Cite journal
| last1 = Bodlaender | first1 = Hans L. | author1-link = Hans L. Bodlaender
| last2 = Thilikos | first2 = Dimitrios M.
| doi = 10.1006/jagm.1999.1011
| issue = 2
| journal = Journal of Algorithms
| pages = 167–194
| title = Graphs with branchwidth at most three
| volume = 32
| year = 1999
| postscript = <!--None-->}}.</ref>
 
==Construction==
The Wagner graph is a [[cubic graph|cubic]] [[Hamiltonian graph]] and can be defined by the [[LCF notation]] [4]<sup>8</sup>.
 
It can be drawn as a [[ladder graph]] with 4 rungs made cyclic on a topological [[Möbius strip]].
 
==Gallery==
<gallery>
Image:Wagner graph 3COL.svg|The [[chromatic number]] of the Wagner graph is&nbsp;3.
Image:Wagner graph 3color edge.svg|The [[chromatic index]] of the Wagner graph is&nbsp;3.
Image:Möbius_ladder_on_Möbius_strip.svg|The Wagner graph drawn on the Möbius strip.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 15:27, 9 September 2014

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