Abstract
In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then \(n\leq h(D)\leq\lfloor\frac{(n+1)^{2}}{4} \rfloor\), and hence characterize the strongly connected digraphs of order n with hamiltonian number \(\lfloor\frac{(n+1)^{2}}{4} \rfloor\). In addition, we show that for each k with \(4\leq n\leq k\leq\lfloor \frac{(n+1)^{2}}{4} \rfloor\), there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.
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The authors thank the referee for many constructive suggestions.
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Second author is supported in part by the National Science Council under grant NSC 99-2115-M-110 -008 -MY3, National Center of Theoretical Sciences.
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Chang, TP., Tong, LD. The hamiltonian numbers in digraphs. J Comb Optim 25, 694–701 (2013). https://doi.org/10.1007/s10878-012-9512-9
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DOI: https://doi.org/10.1007/s10878-012-9512-9