Embeddings of \(m\)-cycle systems and incomplete \(m\)-cycle systems: \(m\leq 14\)
An \(m\)-cycle on a graph \(G\) is a partition of the edges of \(G\) into cycles of length \(m\). The embedding problem for \(m\)-cycle systems asks for the determination of \(S_m(u)\), the set of integers \(v\) so that any \(m\)-cycle system on \(K_u\) can be embedded in an \(m\)-cycle system on \(K_v\). In this paper the authors determine \(S_m(u)\) for \(m \in \{4, 6, 8, 10, 11, 12, 13, 14\}\); \(S_m(u)\) for the other values of \(m \leq 14\) being determined elsewhere. An \(m\)-cycle embedding yields an \(m\)-cycle system on the graph \(K_v - K_u\). However, it is possible to have an \(m\)-cycle system on the graph \(K_v - K_u\) even though there is no \(m\)-cycle system on \(K_u\). The set of values \(u\) and \(v\) for which this is possible is determined for \(m \in \{4, 6, 7, 8, 10, 12, 14\}\).
scientific article; zbMATH DE number 1047079
scientific article