# Definition:Cycle (Graph Theory)

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## Definition

A **cycle** is a circuit in which no vertex except the first (which is also the last) appears more than once.

An **$n$-cycle** is a **cycle** with $n$ vertices.

### Subgraph

The set of vertices and edges which go to make up a **cycle** form a subgraph.

This subgraph itself is also referred to as a **cycle**.

### Odd Cycle

An **odd cycle** is a cycle with odd length, that is, with an odd number of edges.

### Even Cycle

An **even cycle** is a cycle with even length, that is, with an even number of edges.

## Also defined as

Some sources specify a **cycle** as having at least one edge.

Some sources specify that a **cycle** must indeed have at least $3$ edges, presupposing that the graph in which it is embedded is by definition a simple graph.

## Also known as

Some sources refer to a **cycle** as a **closed path**.

## Also see

- Results about
**cycles**in the context of**graph theory**can be found**here**.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 2.3$: Connected Graphs - 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.2$ Graphs and Trees - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**path**:**1.** - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 2.3.4.1$: Free Trees - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**cycle** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**walk** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**cycle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**walk** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**closed**(in graph theory) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cycle**(in graph theory)