Hamiltonian numbers in oriented graphs

A hamiltonian walk of a digraph is a closed spanning directed walk with minimum length in the digraph. The length of a hamiltonian walk in a digraph D is called the hamiltonian number of D, denoted by h(D). In Chang and Tong (J Comb Optim 25:694–701, 2013), Chang and Tong proved that for a strongly connected digraph D of order n, (nle h(D)le lfloor frac{(n+1)^2}{4} rfloor ), and characterized the strongly connected digraphs of order n with hamiltonian number (lfloor frac{(n+1)^2}{4} rfloor ). In the paper, we characterized the strongly connected digraphs of order n with hamiltonian number (lfloor frac{(n+1)^2}{4} rfloor -1) and show that for any triple of integers n, k and t with (nge 5), (nge kge 3) and (tge 0), there is a class of nonisomorphic digraphs with order n and hamiltonian number (n(n-k+1)-t).