# cycle

A *cycle* in a graph, digraph^{}, or multigraph^{}, is a simple path from a vertex to itself (i.e., a path where the first vertex is the same as the last vertex and no edge is repeated).

For example, consider this graph:

$$\text{xymatrix}A\text{ar}\mathrm{@}-[r]\text{ar}\mathrm{@}-[d]\mathrm{\&}B\text{ar}\mathrm{@}-[dl]\text{ar}\mathrm{@}-[d]D\text{ar}\mathrm{@}-[r]\mathrm{\&}C$$ |

$ABCDA$ and $BDAB$ are two of the cycles in this graph. $ABA$ is not a cycle, however, since it uses the edge connecting $A$ and $B$ twice. $ABCD$ is not a cycle because it begins on $A$ but ends on $D$.

A cycle of length $n$ is sometimes denoted ${C}_{n}$ and may be referred to as a polygon^{} of $n$ sides: that is, ${C}_{3}$ is a triangle, ${C}_{4}$ is a quadrilateral^{}, ${C}_{5}$ is a pentagon^{}, etc.

An *even* cycle is one of even length; similarly, an *odd* cycle is one of odd length.

Title | cycle |
---|---|

Canonical name | Cycle |

Date of creation | 2013-03-22 12:17:21 |

Last modified on | 2013-03-22 12:17:21 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 8 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 05C38 |

Related topic | AcyclicGraph |

Related topic | SimplePath |

Related topic | VeblensTheorem |

Related topic | MantelsTheorem |

Related topic | Graph |