Abstract
A hamiltonian walk in a digraph D is a closed spanning directed walk of D with minimum length. The length of a hamiltonian walk in D is called the hamiltonian number of D, and is denoted by h(D). The hamiltonian spectrum \(S_h(G)\) of a graph G is the set \(\{h(D): D\) is a strongly connected orientation of \(G\}\). In this paper, we present necessary and sufficient conditions for a graph G of order n to have \(S_h(G)=\{n\}\), \(\{n+1\}\), or \(\{n+2\}\). Then we construct some 2-connected graphs of order n with hamiltonian spectrum being a singleton \(n+k\) for some \(k\ge 3\), and graphs with their hamiltonian spectra being sets of consecutive integers.
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We thank the referee for a careful reading of the manuscript and valuable comments.
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Li-Da Tong: Supported in part by the National Science Council under grant MOST 105-2115-M-110 -003 -MY2.
Xuding Zhu: Grant Numbers: NSFC 11571319, ZJNSF D19A010003.
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Tong, LD., Yang, HY. & Zhu, X. Hamiltonian Spectra of Graphs. Graphs and Combinatorics 35, 827–836 (2019). https://doi.org/10.1007/s00373-019-02035-0
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DOI: https://doi.org/10.1007/s00373-019-02035-0