The manuscript considers mathematical models for creating a topological drawing of a graph based on the methods of G. Ringel's vertex rotation theory. An algorithm is presented for generating a topological drawing of a flat part of a graph based on the selection of a basis for the cycle subspace C(G) using the Monte Carlo method. A steepest descent method for constructing a topological drawing of a flat subgraph is described in the manuscript. The topological drawing of a graph is constructed using a combination of the methods of vector intersection algebra developed by L. I. Rapport. Three stages of constructing a flat subgraph of a non-separable graph are described. The issues of constructing a Hamiltonian cycle based on constructing a flat subgraph are considered. A new method for constructing a Hamiltonian cycle of a graph based on the cycle graph of a flat subgraph is described.