A walk, a trail or a path (x₀,e₁,x₁,...,e_k, x_k) is said to be closed if its ends x₀ and x_k coincide.

A cycle is a closed path of length ≥ 1, that is a path of the form:

where k ≥ 1, and vertices x_i, for i =0,...,k − 1, are all distinct. Integer k is the length of the cycle. Unlike a walk, and a trail, a cycle cannot have

zero length. The minimum length it can be is 1. In this case, it is made up of one vertex with a loop. When G is a simple graph, a cycle may be defined

by the sequence (x₀,x₁,...,x₀) of its vertices. In this case the length is then umber of vertices of the cycle (except the last one). A cycle is called even

or odd, depending on whether its length is even or odd.

Figure 1.1 gives some examples of walks, trails, paths and cycles.

                                                              Figure 1.1

Note. No distinction is made between cycles that only differ in the cyclic sequences of vertices which define them. For example, in a simple graph, the

three following cycles are considered as one unique cycle:

In fact, in the graph, it is the same cycle described differently.

Graph Theory  and Applications ,Jean-Claude Fournier, WILEY, page(31-33)