A new constraint of the Hamilton cycle algorithm

Grinberg’s theorem is a necessary condition for the planar Hamilton graphs. In this paper, we use cycle bases and removable cycles to survey cycle structures of the Hamiltonian graphs and derive an equation of the interior faces in Grinberg’s Theorem. The result shows that Grinberg’s Theorem is suitable for the connected and simple graphs. Furthermore, by adding a new constraint of solutions to the equation, we find such solutions can be a necessary and sufficient condition for finding Hamiltonian graphs. We use the new constraint to improve the edge pruning technique and obtain a polynomial time algorithm for finding a Hamiltonian cycle in the connected and simple graphs.