Uniformly Resolvable Designs

Abstract

A $k$-URD$(v,g,r)$ is a resolvable design on $v$ points with block sizes $g$ and $k$. Each parallel class contains only one block size, and there are $r$ parallel classes with blocks of size $g$, this implies there are $\frac{v-1-r(g-1)}{k-1}$ parallel classes of size $k$.We show that for sufficiently large $v$, the necessary conditions are sufficient for the following range of values of $r$. Let ${ϵ}_{k,g}=1$if $g\equiv 0\mathrm{mod}k$ and $k$ otherwise, and let $u=\frac{v}{g{ϵ}_{k,g}}$.If $k=2$ for all $g$, or $k=3$ with $g$ odd, then there exists a $k$-URD$(v,g,r)$ for the following values of $r$:

1. If $u$ is an odd prime power, then for all $1\le r\le \frac{1}{k-1}(u-1)$, except possibly for the case where $k=3$ and $u$ is congruent to $5\mathrm{mod}6$.
2. If $u$ is not prime, then for all $1\le r\le \frac{u}{p}$, where $p$ is the smallest proper factor of $u$.