Topologized Hamiltonian and Complete Graph

Topological graph theory deals with embedding the graphs in Surfaces, and the graphs considered as a topological spaces. The concept topology extended to the topologized graph by the S1 space and the boundary of every vertex and edge. The space is S1 if every singleton in the topological space is either open or closed. Let G be a graph with n vertices and e edges and a topology defined on graph is called topologized graph if it satisfies the following:

• Every singleton is open or closed and
• For all x ∈ X, | ∂(x) |≤ 2, where ∂(x) is the boundary of a point x.

This paper examines some results about the topological approach of the Complete Graph, Path, Circuit, Hamiltonian circuit and Hamiltonian path. And the results were generalized through this work.