Difference Necklaces

An (a,b)-difference necklace of length n is a circular arrangement of the integers 0, 1, 2, … , 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.