Enumerating the Walecki‐Type Hamiltonian Cycle Systems |
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| Authors: | Emanuele Brugnoli |
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| Institution: | Department of Mathematics and Computer Sciences, University of Palermo, Palermo, Italy |
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| Abstract: | Let  be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly  ) is a set of Hamiltonian cycles of  whose edges partition the edge set of  . By means of a slight modification of the famous  of Walecki, we obtain 2n pairwise distinct  and we enumerate them up to isomorphism proving that this is equivalent to count the number of binary bracelets of length n, i.e. the orbits of  , the dihedral group of order 2n, acting on binary n‐tuples. |
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| Keywords: | Hamiltonian cycle systems dihedral group integer compositions binary bracelets |
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be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly 
) is a set of Hamiltonian cycles of 
whose edges partition the edge set of 
. By means of a slight modification of the famous 
of Walecki, we obtain 2n pairwise distinct 
and we enumerate them up to isomorphism proving that this is equivalent to count the number of binary bracelets of length n, i.e. the orbits of 
, the dihedral group of order 2n, acting on binary n‐tuples.