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

$ABCDA$ 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.