Abstract
LetT be a hamiltonian tournament withn vertices and γ a hamiltonian cycle ofT. For a cycleC k of lengthk inT we denoteI γ(C k, the number of arcs that γ andC k have in common. Letf(n,k,T,γ)=max{I γ(C k)|C k ⊂T} andf(n, k)=min{f(n, k, T, γ)|T is a hamiltonian tournament withn vertices, and γ a hamiltonian cycle ofT}. In a previous paper [3] we studied the case ofn≥2k−4 and proved that
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.f(n, 3)=1, f(n, 4)=1 andf(n, 5)=2 ifn≠2k−2;
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.f(n, k)=k−1 if and only ifn=2k−2;
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. fork>5,f(n, k)=k−2 if and only ifn≥2k−4,n≠2k−2 andn≡k (modk−2);
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. fork>5,f(n, k)=k−3 if and only ifn≥2k−4 andn≢k (modk−2).
In this paper we consider the case ofn≤2k−5 and complete the description off(n, k) by proving thatf(n, k)=k−4 if and only ifn≤2k−5.
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Part of this work was done at the MIT Laboratory for Computer Science, and the DEC Cambridge Research Laboratory.
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Galeana-Sánchez, H., Rajsbaum, S. Cycle-pancyclism in tournaments III. Graphs and Combinatorics 13, 57–63 (1997). https://doi.org/10.1007/BF01202236
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DOI: https://doi.org/10.1007/BF01202236