On Alspach's conjecture with two even cycle lengths (Q1587593)

Alspach's conjecture states that if \(\pi\) is a partition of the number \(v(v-1)/2\), for \(v\) odd [of the number \(v(v-2)/2\), for \(v\) even] into \(s\) parts, each at least 3 and at most \(v\), then there exists a decomposition of the complete graph \(K_v\) [of the cocktail-party graph \(K_v-F\)] into \(s\) cycles whose lengths equal the parts of the partition \(\pi\). In this paper, the authors deal with the case when all parts are equal to one of two even numbers \(m\), \(n\), \(m< n\). They show that in this case Alspach's conjecture is true if and only if it is true for all \(v< 7n\). This, together with settling several small cases, makes it possible to show that the Alspach conjecture is true for all even \(m\), \(n\leq 10\).