Decompositions of the Complete Graph into Small 2-Regular Graphs

A \(G\)-decomposition of the complete graph \(K_v\) is a set \({S}\) of subgraphs of \(K_v\), each isomorphic to \(G\), such that the edge set of \(K_v\) is partitioned by the edge sets of the subgraphs in \({S}\). For all positive integers \(v\) and every 2-regular graph \(G\) with ten or fewer vertices, we prove necessary and sufficient conditions for the existence of a \(G\)-decomposition of \(K_v\).