The sufficient condition is improved for identifying more edges in optimal Hamiltonian cycles based on frequency quadrilaterals. If the frequency 5s related to an edge occupy more than two-thirds with respect to all frequency quadrilaterals containing it and the others are frequency 3s, this edge is in an optimal Hamiltonian cycle. We also proved that if an edge is contained in an optimal Hamiltonian cycle of one graph, it will be contained in an optimal Hamiltonian cycle of a second graph with a probability bigger than two-thirds as the second graph is expanded from the first graph by adding one vertex. The experiments are conducted to approve the findings.