Non-Hamiltonian Holes in Grid Graphs

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in the basis, we deduce the formula of inside faces in Grinberg theorem, called Grinberg equation, and derive a kind of cycles whose existence make a polygonal grid graph non-Hamiltonian, called non-Hamiltonian holes, and then we characterize the existence condition of non-Hamiltonian holes and obtain the necessary and sufficient condition of a polygonal grid graph to be Hamiltonian. The result in this paper provides a new starting point for developing a polynomial-time algorithm for Hamilton problem in general grid graphs.