Extended Skolem type difference sets

A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples {(di,1, di,2, di,3, di,4, di,5) | i = 1, 2, . . . , t} such that di,1+di,2+di,3+di,4+di,5 = 0 for 1 ≤ i ≤ t and {|di,j| | 1 ≤ i ≤ t, 1 ≤ j ≤ 5} = {1, 2, . . . , 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set of order t. We also consider hooked k-extended Skolem-type 5-tuple difference sets of order t and provide necessary and sufficient conditions for their existence. We will then show how these k-extended Skolem-type difference sets can be used to find decompositions of circulant and complete graphs of order n into 5-cycles, d-cycles, where d is a divisor of n, Hamilton cycles, and possibly a 1-factor.