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 $\mathsf{H}\mathsf{a}\mathsf{m}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a contractad. Similarly, we construct the graphical collection of Hamiltonian cycles $\mathsf{C}\mathsf{y}\mathsf{c}\mathsf{H}\mathsf{a}\mathsf{m}$ that forms a right module over the contractad $\mathsf{H}\mathsf{a}\mathsf{m}$. We use the machinery of contractad generating series for counting Hamiltonian paths/cycles for particular types of graphs.