A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples In this paper, we 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 then show how these k-extended Skolemtype difference sets can be used to find decompositions of circulant and

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... plete graphs of order n into 5-cycles, d-cycles, where d is a divisor of n, Hamilton cycles, and possibly a 1-factor.