Determination of the size of defining set for Steiner triple systems

TL;DR

This paper investigates the size of defining sets for 3-colorings of Steiner triple systems, determining exact values for small systems and establishing bounds for larger ones.

Contribution

It provides the first comprehensive analysis of defining sets in Steiner triple systems, including exact sizes for systems up to order 15 and bounds for larger systems.

Findings

Determined minimum defining sets for all non-isomorphic STS(v) with v ≤ 15.

Calculated the defining number for all Steiner triple systems of order v.

Established lower bounds for the size of the largest minimal defining set for all STS(v).

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 $v$.