Resolving Triple Systems into Regular Configurations

Abstract

A $\lambda-Triple\ System(v)$, or a $\lambda $–$TS(V,\cal{B})$, is a pair (V, $\cal{B}$) where V is a set and $\cal{B}$ is a subset of the 3-subsets of V so that every pair is in exactly $\lambda$ elements of $\cal{B}$. A $regular\ configuration$ on p points with regularity $\rho$ on $l$ blocks is a pair (P,${\cal L}$) where $\cal{L}$ is a collection of 3-subsets of a (usually small) set P so that every p in P is in exactly $\rho$ elements of ${\cal L}$, and $|{\cal L}|=l$. The Pasch configuration $(\{0,1,2,3,4,5\},\{ 012,035,245,134\})$ has p=6, $l$=4, and $\rho$=2. A $\lambda$–$TS(V,\cal{B})$, is resolvable into a regular configuration ${\Bbb C}$=(P,${\cal L}$), or ${\Bbb C}$–resolvable, if ${\cal B}$ can be partitioned into sets $\Pi_{i}$ so that for each i, (V,$\Pi_{i}$) is isomorphic to a set of vertex disjoint copies of (P,${\cal L}$). If the configuration is a single block on three points this corresponds to ordinary resolvability of a Triple System.

In this paper we examine all regular configurations ${\Bbb C}$ on 6 or fewer blocks and construct ${\Bbb C}$–resolvable $\lambda$–Triple Systems of order v for many values of v and $\lambda$. These conditions are also sufficient for each ${\Bbb C}$ having 4 blocks or fewer. For example for the Pasch configuration $\lambda \equiv 0 \pmod{4}$ and $v \equiv 0 \pmod{6}$ are necessary and sufficient.