On Nesting of Path Designs

Let \(h \geq 1\). For each admissible \(v\), we exhibit a nested balanced path design \(H(v, 2h+1, 1)\). For each admissible odd \(v\), we exhibit a nested balanced path design \(H(v,2h,1)\). For every \(v \equiv 4 \pmod{6}\), \(v \geq 10\), we exhibit a nested balanced path design \(H(v,4,1)\) except possibly if \(v \in \{16, 52, 70\}\).

For each \(v \equiv 0 \pmod{4h}\), \(v \geq 4h\), we exhibit a nested path design \(P(v,2h+1,1)\). For each \(v \equiv 0 \pmod{4h-2}\), \(v \geq 4h-2\), we exhibit a nested path design \(P(v,2h,1)\). For every \(v \equiv 3 \pmod{6}\), \(v \geq 9\), we exhibit a nested path design \(P(v,4,1)\) except possibly if \(v = 39\).