A Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once… Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. - wikipedia

To be compared with Eulerian path, a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices).

Algorithm ?

Finding them is trivial, but very timely if done via DFS. When considering Topcoder’s 2 second timeout, brutal for this problem, we have to resort to another solution, the Held-Karp Algorithm implemented with a technique called bitDP. - Hamiltonian Paths & bitDP