Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian Cycles

Let G = (V, E) be a bipartite graph embedded in a plane (or n-holed torus). Two subgraphs of G differ by a Z-transformation if their symmetric difference consists of the boundary edges of a single face—and if each subgraph contains an alternating set of the edges of that face. For a given φ : V 7→ Z, Sφ is the set of subgraphs of G in which each v ∈ V has degree φ(v). Two elements of Sφ are said to be adjacent if they differ by a Z-transformation. We determine the connected components of Sφ and assign a height function to each of its elements. If φ is identically two, and G is a grid graph, Sφ contains the partitions of the vertices of G into cycles. We prove that we can always apply a series of Z-transformations to decrease the total number of cycles provided there is enough “slack” in the corresponding height function. This allows us to determine in polynomial time the minimal number of cycles into which G can be partitioned provided G has a limited number of non-square faces. In particular, we determine the Hamiltonicity of polyomino graphs in polynomial time, thereby solving a problem posed by Itai, Christos, and Szwarcfiter in 1982 [21]. The algorithm extends to n-holed-torusembedded graphs that have grid-like properties. We also provide Markov chains for sampling and approximately counting the Hamiltonian cycles