Ingraph theory, a cycle graphorcircular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn.[2] The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.
| Cycle graph | |
|---|---|
The cycle graph C5
| |
| Girth | n |
| Automorphisms | 2n (Dn) |
| Chromatic number | 3 if n is odd 2 otherwise |
| Chromatic index | 3 if n is odd 2 otherwise |
| Spectrum |  [1] |
| Properties | 2-regular Vertex-transitive Edge-transitive Unit distance Hamiltonian Eulerian |
| Notation | Cn |
| Table of graphs and parameters | |
There are many synonyms 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 acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used. The term n-cycle is sometimes used in other settings.[3]
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.
A cycle graph is:
In addition:
Similarly to the Platonic graphs, the cycle graphs form the skeletons of the dihedra. Their duals are the dipole graphs, which form the skeletons of the hosohedra.
Adirected 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 (orarc) 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 1 and uniform out-degree 1.
Directed cycle graphs are Cayley graphs for cyclic groups (see e.g. Trevisan).
