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.
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