Hamiltonian numbers in oriented graphs

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 $n(n-k+1)-t$.