Operadic structure on Hamiltonian paths and cycles

We study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection Ham encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles CycHam that forms a right module over the contractad Ham. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.