Directed Covering with Block Size 5 and $v$ and $\lambda$ Odd

Abstract

A directed covering design, $DC(v,k,\lambda )$, is a $(v,k,2\lambda )$ covering design in which the blocks are regarded as ordered $k$-tuples and in which each ordered pair of elements occurs in at least $\lambda$ blocks. Let $DE(v,k,\lambda )$ denote the minimum number of blocks in a $DC(v,k,\lambda )$. In this paper, the values of the function $DE(v,k,\lambda )$ are determined for all odd integers $v\ge 5$ and $\lambda$ odd, with the exception of $(v,\lambda )=(53,1),(63,1),(73,1),(83,1)$. Further, we provide an example of a covering design that cannot be directed.