Cycle graph

 ( Parametric Families Of Graphs ) 

In graph theory, a cycle graph or circular 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 graph) connected in a closed chain. The cycle graph with vertices is called . p. 8, §1.3 The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.

Cycle graph $C\_1$ is an isolated loop. Cycle graph $C\_3$ is the same as Complete graph $K\_3$.

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Terminology There are many 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.

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.

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Properties A cycle graph is:

• k-edge colorable, if and only if it has an even number of vertices
• regular graph
• Bipartite graph, 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 (Kőnig, 1936).
• Connected graph
• Eulerian graph
• Hamiltonian
• A unit distance graph

In addition:

• As cycle graphs can be graph drawing as , the 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.

Similarly to the , the cycle graphs form the skeletons of the dihedron. Their duals are the , which form the skeletons of the hosohedron.

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Directed cycle graph 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 1 and uniform out-degree 1.

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

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See also

• Complete bipartite graph
• Complete graph
• Circulant graph
• Cycle graph (algebra)
• Null graph
• Path graph

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Sources

• Reinhard Diestel (2026). 9783662536216, Springer. ISBN 9783662536216

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External links

• (discussion of both 2-regular cycle graphs and the group-theoretic concept of )
• Luca Trevisan, Characters and Expansion.

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