On Nesting of Path Designs

Abstract

Let $h\ge 1$. 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 $v\equiv 4(\mathrm{mod}6)$, $v\ge 10$, we exhibit a nested balanced path design $H(v,4,1)$ except possibly if $v\in \{16,52,70\}$.

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