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Enumerating the Walecki‐Type Hamiltonian Cycle Systems
Author(s) -
Brugnoli Emanuele
Publication year - 2017
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21558
Subject(s) - mathematics , combinatorics , dihedral group , hamiltonian path , pairwise comparison , hamiltonian (control theory) , isomorphism (crystallography) , partition (number theory) , graph isomorphism , binary number , tuple , discrete mathematics , graph , group (periodic table) , crystallography , arithmetic , line graph , mathematical optimization , chemistry , statistics , organic chemistry , crystal structure
Let K v be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly HCS ( v ) ) is a set of Hamiltonian cycles of K v whose edges partition the edge set of K v . By means of a slight modification of the famous HCS ( 4 n + 1 ) of Walecki, we obtain 2 n pairwise distinct HCS ( 4 n + 1 ) 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 D n , the dihedral group of order 2 n , acting on binary n ‐tuples.