Balanced Transitive Orientations

Abstract

A transitive orientation of a partial triple system $(S,T)$ of index $2\lambda$ is a partial transitive triple system formed by replacing each triple $t\in T$ with a transitive triple defined on the same vertex set as $t$, such that each ordered pair occurs in at most $\lambda$ of the resulting transitive triples. A transitive orientation $(S_1,T_1)$ of $(S,T)$ is said to be balanced if for all $\{u,v\}\subseteq S$, if $\{u,v\}$ occurs in $\ell$ triples in $T$ then $\lfloor \ell /2\rfloor$
and $\lceil \ell /2\rceil$ transitive triples in $T_1$ contain the arcs $(u,v)$ and $(v,u)$ respectively. In this paper, it is shown that every partial triple
system has a balanced transitive orientation. This result is then used to prove the existence of certain transitive group divisible designs.