a path connecting all of the vertices, but traverses through any one vertex only once

a path that passes through each vertex exactly once

A hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A hamiltonian path does not necessarily pass through all the edges of the graph, however. A hamiltonian path which ends in the same place in which it began is called a hamiltonian circuit or a hamiltonain cycle.

A cycle in a directed or undirected graph such that every vertex of the graph is in the cycle, but each vertex appears exactly once (except the first and last vertices).

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph which visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected graph which visits each vertex exactly once and also returns to the starting vertex. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem which is NP-complete.