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{{About|connected, 2-regular graphs}}
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{{infobox graph
 | name = Cycle graph
 | image = [[Image:Undirected 6 cycle.svg|160px]]
 | image_caption = A cycle graph of length 6
 | vertices = ''n''
 | edges = ''n''
 | automorphisms    = 2''n'' (''D<sub>n</sub>'')
 | chromatic_number = 3 if ''n'' is odd<br/>2 if ''n'' is even
 | chromatic_index = 3 if ''n'' is odd<br/>2 if ''n'' is even
 | girth = ''n''
 | spectrum = {2 cos(2 ''k'' π / ''n''); ''k''=1, ... ,''n''}<ref>[http://www.win.tue.nl/~aeb/2WF02/easyspectra.pdf Some simple graph spectra]. win.tue.nl</ref>
 | notation = <math>C_n</math> 
 | properties = [[Regular graph|2-regular]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[Edge-transitive graph|Edge-transitive]]<br>[[Unit distance graph|Unit distance]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Eulerian graph|Eulerian]]
}}
In [[graph theory]], a '''cycle graph''' or '''circular graph''' is a [[graph (mathematics)|graph]] that consists of a single [[Cycle (graph theory)|cycle]], or in other words, some number of vertices connected in a closed chain. The cycle graph with ''n'' vertices is called ''C<sub>n</sub>''. The number of vertices in ''C<sub>n</sub>'' equals the number of [[Edge (graph theory)|edge]]s, and every vertex has [[degree (graph theory)|degree]]&nbsp;2; that is, every vertex has exactly two edges incident with it.

==Terminology==
There are many [[synonym]]s for "cycle graph". These include '''simple cycle graph''' and '''cyclic graph''', although the latter term is less often used, because it can also refer to graphs which are merely not [[directed acyclic graph|acyclic]]. Among graph theorists, '''cycle''', '''polygon''', or '''''n''-gon''' are also often used. A cycle with an even number of vertices is called an '''even cycle'''; a cycle with an odd number of vertices is called an '''odd cycle'''.

==Properties==
A cycle graph is:
* [[Connected graph|Connected]]
* [[regular graph|2-regular]]
* [[Eulerian graph|Eulerian]]
* [[Hamiltonian graph|Hamiltonian]]
* [[Bipartite graph|2-vertex colorable]], if and only if it has an even number of vertices. More generally, a graph is bipartite [[if and only if]] it has no odd cycles ([[Dénes Kőnig|Kőnig]], 1936).
* [[k-edge colorable|2-edge colorable]], if and only if it has an even number of vertices
* 3-vertex colorable and 3-edge colorable, for any number of vertices
* A [[unit distance graph]]

In addition: 
*As cycle graphs can be [[graph drawing|drawn]] as [[regular polygon]]s, the [[automorphism group|symmetries]] of an ''n''-cycle are the same as those of a regular polygon with ''n'' sides, the [[dihedral group]] of order 2''n''. In particular, there exist symmetries taking any vertex to any other vertex, and any edge to any other edge, so the ''n''-cycle is a [[symmetric graph]].

==Directed cycle graph==
[[Image:DC8.png|frame|right|A directed cycle graph of length 8]]
A '''directed cycle graph''' is a directed version of a cycle graph, with all the edges being oriented in the same direction.

In a [[directed graph]], a set of edges which contains at least one edge (or ''arc'') from each directed cycle is called a [[feedback arc set]]. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a [[feedback vertex set]].

A directed cycle graph has uniform in-degree&nbsp;1 and uniform out-degree&nbsp;1.

Directed cycle graphs are [[Cayley graph]]s for [[cyclic group]]s (see e.g. Trevisan).

==See also==
{{commons category|Cycle graphs}}
* [[Complete bipartite graph]]
* [[Path graph]]
* [[Complete graph]]
* [[Null graph]]

==References==
{{reflist}}

==External links==
*{{MathWorld |urlname=CycleGraph |title=Cycle Graph}} (discussion of both 2-regular cycle graphs and the group-theoretic concept of [[cycle diagram]]s)
*[[Luca Trevisan]], [http://in-theory.blogspot.com/2006/12/characters-and-expansion.html Characters and Expansion].

[[Category:Parametric families of graphs]]
[[Category:Regular graphs]]