Tutte theorem: Difference between revisions

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{{infobox graph
| name = Wheel graph
| image = [[Image:Wheel graphs.svg|220px]]
| image_caption = Several examples of wheel graphs
| girth    = 3
| diameter = 2 if ''n>4''<br>1 if ''n''=4
| vertices = ''n''
| edges = 2(''n'' &minus; 1)
| chromatic_number = 3 if ''n'' is odd<br/>4 if ''n'' is even
| chromatic_index =
| spectrum = <math>\{2 \cos(2 k \pi / (n-1))^1; k = 1, \ldots, n - 2\}</math><math>\cup \{1 \pm \sqrt{n}\}</math>
| properties = [[Hamiltonian graph|Hamiltonian]]<br>[[Dual graph|Self-dual]]<br>[[Planar graph|Planar]]
| notation = ''W''<sub>''n''</sub>
}}


In the [[mathematical]] discipline of [[graph theory]], a '''wheel graph''' ''W''<sub>''n''</sub> is a graph with ''n'' vertices (n ≥ 4), formed by connecting a single vertex to all vertices of an (''n''-1)-[[Cycle graph|cycle]]. The numerical notation for wheels is used inconsistently in the literature: some authors instead use ''n'' to refer to the length of the cycle, so that their ''W''<sub>''n''</sub> is the graph we denote ''W''<sub>''n+1''</sub>.<ref>{{mathworld | urlname = WheelGraph | title = Wheel Graph}}</ref> A wheel graph can also be defined as the 1-[[Skeleton (topology)|skeleton]] of an (''n''-1)-gonal [[Pyramid (geometry)|pyramid]].


==Set-builder construction==
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Given a vertex set of {1,2,3,…,v}, the edge set of the wheel graph can be represented in [[set-builder notation]] by {{1,2},{1,3},…,{1,v},{2,3},{3,4},…,{v-1,v},{v,2}}.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=56|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|accessdate=8 August 2012}}</ref>
 
==Properties==
Wheel graphs are [[planar graph]]s, and as such have a unique planar embedding. More specifically, every wheel graph is a [[Halin graph]]. They are self-dual: the [[Dual graph|planar dual]] of any wheel graph is an isomorphic graph. Any maximal planar graph, other than ''K''<sub>4</sub> = ''W''<sub>4</sub>, contains as a subgraph either ''W''<sub>5</sub> or ''W''<sub>6</sub>.
 
There is always a [[Hamiltonian cycle]] in the wheel graph and there are <math>n^2-3n+3</math> cycles in ''W''<sub>''n''</sub> {{OEIS|id=A002061}}.
 
{| class=wikitable
|[[Image:CyclesW4.svg|320px]]<BR>The 7 cycles of the wheel graph ''W''<sub>4</sub>.
|}
 
For odd values of ''n'', ''W''<sub>''n''</sub> is a [[perfect graph]] with [[chromatic number]] 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even ''n'', ''W''<sub>''n''</sub> has [[chromatic number]] 4, and (when ''n'' ≥ 6) is not perfect. ''W''<sub>7</sub> is the only wheel graph that is a [[unit distance graph]] in the Euclidean plane.<ref>{{citation
  | last1 = Buckley | first1 = Fred | last2 = Harary | first2 = Frank | author2-link = Frank Harary
  | title = On the euclidean dimension of a wheel
  | journal = Graphs and Combinatorics
  | volume = 4
  | issue = 1
  | year = 1988
  | doi = 10.1007/BF01864150
  | pages = 23–30}}.</ref>  
 
The [[chromatic polynomial]] of the wheel graph ''W<sub>n</sub>'' is :
<center><math>P_{W_n}(x)=x((x-2)^{(n-1)}-(-1)^{n}(x-2)).</math></center>
 
In [[matroid]] theory, two particularly important special classes of matroids are the ''wheel matroids'' and the ''whirl matroids'', both derived from wheel graphs. The ''k''-wheel matroid is the [[graphic matroid]] of a wheel ''W''<sub>''k+1''</sub>, while the ''k''-whirl matroid is derived from the ''k''-wheel by considering the outer cycle of the wheel, as well as all of its [[spanning tree]]s, to be independent.
 
The wheel ''W''<sub>6</sub> supplied a counterexample to a conjecture of [[Paul Erdős]] on [[Ramsey theory]]: he had conjectured that the complete graph has the smallest Ramsey number among all graphs with the same chromatic number, but Faudree and McKay (1993) showed ''W''<sub>6</sub> has Ramsey number 17 while the complete graph with the same chromatic number, ''K''<sub>4</sub>, has Ramsey number 18.<ref>{{citation
  | last1 = Faudree | first1 = Ralph J. | author1-link = Ralph Faudree
  | last2 = McKay | first2 = Brendan D. | author2-link = Brendan McKay
  | title = A conjecture of Erdős and the Ramsey number ''r''(''W''<sub>6</sub>)
  | url = http://cs.anu.edu.au/people/Brendan.McKay/papers/wheels.ps.gz
  | journal = J. Combinatorial Math. and Combinatorial Comput. <!-- comment to prevent DOI bot from breaking the journal name again -->
  | volume = 13
  | year = 1993
  | pages = 23–31}}.</ref> That is, for every 17-vertex graph ''G'', either ''G'' or its complement contains ''W''<sub>6</sub> as a subgraph, while neither the 17-vertex [[Paley graph]] nor its complement contains a copy of ''K''<sub>4</sub>.
 
==References==
{{reflist}}
 
[[Category:Parametric families of graphs]]
[[Category:Planar graphs]]

Latest revision as of 04:15, 10 December 2014


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- A specific comprehension of the syndication course of action

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- A passion for, and devotion to, operating brilliance

- A proven capability to establish a great carrying out business and team

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