Maximum $2\times 3$ Grid-Block Packings of $K_v$

Abstract:

      Let $K_v$ be the complete graph on $v$ vertices, and $G$ a finite simple undirected graph without isolated vertices. A $G$-packing of $K_v$, denoted by $(v,G,1)$-packing, is a pair $(X,\mathcal{A})$ where $X$ is the vertex set of $K_v$ and $\mathcal{A}$ is a family of edge-disjoint subgraphs isomorphic to $G$ in $K_v$. In this paper, the maximum number of subgraphs in a $(v,G,1)$-packing is determined when $G$ is $K_2\times K_3$, the Cartesian product of $K_2$ and $K_3$, leaving two orders undetermined. This design originated from the use of DNA library screening.