For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced the concept of Hamiltonian number. The Hamiltonian number
h(D) of a digraph D is the minimum length of a closed walk containing all vertices of D. In this paper, we study Hamiltonian numbers of the following proposed networks, which include strongly connected double loop networks. For integers d≥1, m≥1 and 𝓁≥0, the Möbius double loop network MDL(d,m,𝓁) is the digraph with vertex set {(i,j):0≤i≤d-1,0≤j≤m-1} and arc set {(i,j)(i+1,j) or (i,j)(i+1,j+1):0≤i≤d-2,0≤j≤m-1}⋃{(d-1,j)(0,j+𝓁) or (d-1,j)(0,j+𝓁+1):0≤j≤m-1}, where the second coordinate y of a vertex (x,y) is taken modulo m. We give an upper bound for the Hamiltonian number of a Möbius double loop network. We also give a necessary and sufficient condition for a Möbius double loop network MDL(d,m,𝓁) to have Hamiltonian number at most dm, dm+d, dm+1 or dm+2.
