Difference Necklaces
Ethan P. White, Richard K. Guy, Renate Scheidler

TL;DR
This paper investigates the existence and enumeration of $(a,b)$-difference necklaces, circular arrangements with specific neighbor differences, providing recurrence relations and generalizing to multiple differences.
Contribution
It establishes existence conditions, derives recurrence relations for specific $(a,b)$ pairs, and generalizes enumeration methods to necklaces with multiple differences.
Findings
Existence conditions for $(a,b)$-difference necklaces.
Explicit recurrence relations for $(1,2)$, $(1,3)$, $(2,3)$, and $(1,4)$.
Generalization to necklaces with multiple allowed differences.
Abstract
An -difference necklace of length is a circular arrangement of the integers such that any two neighbours have absolute difference or . We prove that, subject to certain conditions on and , such arrangements exist, and provide recurrence relations for the number of -difference necklaces for , , and . Using techniques similar to those employed for enumerating Hamiltonian cycles in certain families of graphs, we obtain these explicit recurrence relations and prove that the number of -difference necklaces of length satisfies a linear recurrence relation for all permissible values and . Our methods generalize to necklaces where an arbitrary number of differences is allowed.
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Taxonomy
Topicsgraph theory and CDMA systems · Advanced Combinatorial Mathematics · Genome Rearrangement Algorithms
