Abstract
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, \(n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor \), 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 \(n\ge 5\), \(n\ge k\ge 3\) and \(t\ge 0\), there is a class of nonisomorphic digraphs with order n and hamiltonian number \(n(n-k+1)-t\).
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The authors thank the referee for many valuable and constructive suggestions.
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Supported in part by the National Science Council under Grant MOST 105-2115-M-110 -003 -MY2.
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Tong, LD., Yang, HY. Hamiltonian numbers in oriented graphs. J Comb Optim 34, 1210–1217 (2017). https://doi.org/10.1007/s10878-017-0141-1
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DOI: https://doi.org/10.1007/s10878-017-0141-1