Cycle graph

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 two, 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 appears to be used more often by non-graph theorists. Among graph theorists, cycle, polygon, or n-gon are also often used. More specifically, a cycle with an even number of vertices is called even cycle, a cycle with an odd number of vertices is called odd cycle.

Properties
A cycle graph is:
 * connected
 * 2-regular
 * Eulerian
 * Hamiltonian
 * symmetric
 * 2-vertex colorable and 2-edge colorable if it has an even number vertices
 * 3-vertex colorable and 3-edge colorable if it has an odd number of vertices.
 * A unit distance graph

In addition:
 * Any graph with a subgraph that is a cycle is not a tree.
 * Cycles with an even number of vertices are bipartite, cycles with an odd number are not. More generally, a graph is bipartite if and only if it has no odd cycles as subgraphs.
 * Cycles with an even number of vertices can be decomposed into a minimum of 2 independent sets (that is, $$\alpha(n)=2$$), whereas cycles with an odd number of vertices can be decomposed into a minimum of 3 independent sets (that is, $$\alpha(n)=3$$).

Directed cycle graph
A directed cyclic graph is a directed version of a cyclic graph, with all the edges being oriented in 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.

There is a directed cycle through any two vertices in a strongly connected component.

External link

 * Eric W. Weisstein, Cyclic Graph at MathWorld.

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