Hamiltonian Cycles

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 = {$\langle$G$\rangle$$\mid$ 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 $\neq$ O(n^k) for any constant k.

In fact, HAM-CYCLE is NP-Complete.