Sums and differences along Hamiltonian cycles

Details Sums and differences along Hamiltonian cycles Sums and differences along Hamiltonian cycles

2006-01-26

arxiv.org/abs/math/0601633v3

Mathematics

Combinatorics

Largest possible number of sums along a Hamiltonian cycle ($\sigma_{max}$) determined completely for all finite abelian groups

Scientific paper

Given a finite abelian group $G$, consider the complete graph on the set of all elements of $G$. Find a Hamiltonian cycle in this graph and for each pair of consecutive vertices along the cycle compute their sum. What are the smallest and the largest possible number of sums that can emerge in this way? What is the expected number of sums if the cycle is chosen randomly? How the answers change if an orientation is given to the cycle and differences (instead of sums) are computed? We give complete solutions to some of these problems and establish reasonably sharp estimates for the rest.

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