Decompositions of the Complete Graph into Small $2$-Regular Graphs

Abstract

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 $K_v$.