A construction for regular-graph designs

Abstract

A regular-graph design is a block design for which a pair $\{a,b\}$ of distinct points occurs in $\hat{I}»+1$ or $\hat{I}»$ blocks depending on whether $\{a,b\}$ is or is not an edge of a given $\hat{I}´$-regular graph. Our paper describes a specific construction for regular-graph designs with $\hat{I}»=1$ and block size $\hat{I}´+1$. We show that for $\hat{I}´\in \{2,3\}$, certain necessary conditions for the existence of such a design with $n$ points are sufficient, with two exceptions in each case and two possible exceptions when $\hat{I}´=3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.

18 pages. Some corrections have been made for this version. Reference added