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 \( v \geq 5 \) 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.