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

Heather Jordon

Abstract


In this paper, we settle Alspach's problem in the case of Hamilton cycles and 5-cycles; that is, we show that for all odd integers $n\ge 5$ and all nonnegative integers $h$ and $t$ with $hn + 5t = n(n-1)/2$, the complete graph $K_n$ decomposes into $h$ Hamilton cycles and $t$ 5-cycles and for all even integers $n \ge 6$ and all nonnegative integers $h$ and $t$ with $hn + 5t = n(n-2)/2$, the complete graph $K_n$ 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.


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