Is it necessary for hamiltonian cycle to cover all the vertices of the graph??

Question

I have read the definition on wikipedia and it says:

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle.

This definition does not say that this cycle should cover every vertex of the graph.

Till today I assumed that hamiltonian cycle should cover all the vertices But,today my teacher was reducing the hamiltonian cycle to Travelling salesman problem and he told that this condition is not necessary Can Anyone please let me know the correct definition as well as any source of it's definition.

Answer 1

Yes, of course. Otherwise it's just a cycle (or circuit, as some would prefer to say.) The definition you quote does include this.

Answer 2

Note that a Hamiltonian cycle is not a graph, it is a cycle (set of edges). Thus, a Hamiltonian cycle doesn't have any vertices, it may only visit them but not contain. Thus, "each vertex" here means each vertex of the graph which contains a Hamiltonian cycle.

So, yes, a Hamiltonian cycle of the graph G visits all of the vertices of G by definition.