Alspach's Problem: The Case of Hamilton Cycles and 5-Cycles

In this paper, we settle Alspach’s problem in the case of Hamilton cycles and 5cycles; that is, we show that for all odd integers n ≥ 5 and all nonnegative integers h and t with hn + 5t = n(n − 1)/2, the complete graph Kn decomposes into h Hamilton cycles and t 5-cycles and for all even integers n ≥ 6 and all nonnegative integers h and t with hn+5t = n(n−2)/2, the complete graph Kn decomposes into h Hamilton cycles, t 5-cycles, and a 1-factor. We also settle Alspach’s problem in the case of Hamilton cycles and 4-cycles.