Determination of the size of defining set for Steiner triple systems

Abstract

Every Steiner triple system is a uniform hypergraph. The coloring of hypergraph and its special case Steiner triple systems, {STS}$(v)$, is studied extensively. But the defining set of the coloring of hypergraph even its special case {STS}$(v)$, is not explored yet. We study minimum defining set and the largest minimal defining set for $3$-coloring of {STS}$(v)$. We determined minimum defining set and the largest minimal defining set, for all non-isomorphic {STS}$(v)$, $v\le 15$. Also we have found the {\sf defining number} for all Steiner triple systems of order $v$, and some lower bounds for the size of the largest minimal defining set for all Steiner triple systems of order $v$, for each admissible $v$.