A Construction for Regular‐Graph Designs

A regular‐graph design is a block design for which a pair a b{ , } of distinct points occurs in λ + 1 or λ blocks depending onwhether a b{ , } is or is not an edge of a given δ‐regular graph. Our paper describes a specific construction for regular‐graphdesigns with λ = 1 and block size δ + 1. We show that for ∈δ {2, 3}, certain necessary conditions for the existence of such adesign with n points are sufficient, with two exceptions in each case and two possible exceptions when δ = 3. We also constructdesigns of orders 105 and 117 for connected 4‐regular graphs.