Determination of the size of defining set for Steiner triple systems

Abstract

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), has not been 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 £ 15. Also we have found the 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.

Subject Classification:

• 05C65
• 05B05
• 05C15

Keywords:

• Hypergraph
• Steiner triple systems
• Coloring
• Defining set