A Construction for Regular-Graph Designs

A regular-graph design is a block design for which a pair $\left\{a,b\right\}$ of distinct points occurs in $\lambda +1$ or $\lambda$ blocks depending on whether $\left\{a,b\right\}$ is or is not an edge of a given $\delta$-regular graph. Our paper describes a specific construction for regular-graph designs with $\lambda =1$ and block size $\delta +1$. We show that for $\delta \in \left\{2,3\right\}$, 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 $\delta =3$. We also construct designs of orders 105 and 117 for connected 4-regular graphs.