Ovals in handcuffed designs of order \(v\) and block size 3 (Q686661)

A handcuffed design \(H(v,3,1)\) is a pair \((V,B)\), where \(V\) is the vertex set of \(K_ v\), the complete graph on \(v\) points, and \(B\) is an edge- disjoint decomposition of \(K_ v\) into copies of \(P_ 3\), the path of length three, such that each vertex belongs to exactly \(r\;P_ 3\)'s. An oval in \(H(v,3,1)\) is a subset of \(V\) with precisely one tangent block at each point and containing no block. The author determines the possible sizes for the ovals and constructs, in each case, a handcuffed design \(H(v,3,1)\) containing an oval. Furthermore, he exhibits, for every admissible \(v>5\), an oval-free \(H(v,3,1)\).