Difference Necklaces

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 $(a,b)$-difference necklace of length $n$ is a circular arrangement of the integers $0, 1, 2, \ldots , n-1$ such that any two neighbours have absolute difference $a$ or $b$. We prove that, subject to certain conditions on $a$ and $b$, such arrangements exist, and provide recurrence relations for the number of $(a,b)$-difference necklaces for $( a, b ) = ( 1, 2 )$, $( 1, 3 )$, $( 2, 3 )$ and $( 1, 4 )$. 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 $(a,b)$-difference necklaces of length $n$ satisfies a linear recurrence relation for all permissible values $a$ and $b$. Our methods generalize to necklaces where an arbitrary number of differences is allowed.