A **Hamiltonian Cycle** of an undirected graph G = (V, E) is a simple cycle
that contains each vertex in V.

Hamiltonian Cycle Decision Problem: Does a graph G have a Hamiltonian Cycle?

Language: HAM-CYCLE = {G G contains a Hamiltonian Cycle}

**Naive Solution:** Try all possible cycles.

If encode graph as an adjacency matrix and n =
G
, then the number of vertices m in G is
. There are m! permutations of vertices (possible
cycles); thus, running time is
=
=
, which is O(*n*^{k}) for any constant k.

In fact, HAM-CYCLE is NP-Complete.