Hamiltonian numbers in oriented graphs

Li-Da Tong () and Hao-Yu Yang
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Li-Da Tong: National Sun Yat-sen University
Hao-Yu Yang: National Sun Yat-sen University

Journal of Combinatorial Optimization, 2017, vol. 34, issue 4, No 14, 1210-1217

Abstract: 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, $$n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor $$ n ≤ h ( D ) ≤ ⌊ ( n + 1 ) 2 4 ⌋ , and characterized the strongly connected digraphs of order n with hamiltonian number $$\lfloor \frac{(n+1)^2}{4} \rfloor $$ ⌊ ( n + 1 ) 2 4 ⌋ . In the paper, we characterized the strongly connected digraphs of order n with hamiltonian number $$\lfloor \frac{(n+1)^2}{4} \rfloor -1$$ ⌊ ( n + 1 ) 2 4 ⌋ - 1 and show that for any triple of integers n, k and t with $$n\ge 5$$ n ≥ 5 , $$n\ge k\ge 3$$ n ≥ k ≥ 3 and $$t\ge 0$$ t ≥ 0 , there is a class of nonisomorphic digraphs with order n and hamiltonian number $$n(n-k+1)-t$$ n ( n - k + 1 ) - t .

Keywords: Hamiltonian number; Digraph; Orientation; 05C45; 05C38; 05C69 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-017-0141-1

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