In graph theory, a cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called C_n. The number of vertices in a C_n equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.

A note on terminology

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. 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
• 2-regular
• Eulerian
• Hamiltonian
• 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 (K?nig, 1936).
• 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 drawn as regular polygons, the symmetries of an n-cycle are the same as those of a regular polygon with n sides, the dihedral group of order 2n. 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

A directed cycle graph of length 8

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 1 and uniform out-degree 1.

Directed cycle graphs are Cayley graphs for cyclic groups (see e.g. Trevisan).

See also

• Complete bipartite graph
• Path graph
• Complete graph
• Null graph

External links

• Eric W. Weisstein, Cycle Graph at MathWorld. (MathWorld discusses both 2-regular cycle graphs and the group-theoretic concept of cycle diagrams in the same article.)

• Luca Trevisan, Characters and Expansion.

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