A Conjecture on the Relation between Three Types of Oriented Triple Systems

Abstract:

      A Mendelsohn (directed, or hybrid) triple system of order $v$, denoted by $\MTS(v, \lambda)$ $(\DTS(v, \lambda)$, or $\HTS(v,\lambda))$, is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}$ is a collection of some cyclic (transitive, or cyclic and transitive) triples on $X$ such that every ordered pair of $X$ belongs to $\lambda$ triples of ${\cal B}$. In this paper, a relation between three types of oriented triple systems was discussed. We conjecture: the block-incident graph of $\MTS(v,\lambda)$ is 3-edge colorable. Then we obtain three disjoint $\DTS(v,\lambda)$s and four disjoint $\HTS(v,\lambda)$s from a given $\MTS(v,\lambda)$.