Representing graphs in Steiner triple systems - II

Let G = (V,E) be a simple graph and let T = (P,B) be a Steiner triple system. Let φ be a one-to-one function from V to P . Any edge e = {u, v} has its image {φ(u), φ(v)} in a unique block in B. We also denote this induced function from edges to blocks by φ. We say that T represents G if there exists a one-to-one function φ : V → P such that the induced function φ : E → B is also one-to-one; that is, if we can represent vertices of the graph by points of the triple system such that no two edges are represented by the same block. The concept was introduced in a previous paper [Graphs Combin. 30 (2014), 255–266], where various results were proved. When the graph to D. ARCHDEACON ET AL. /AUSTRALAS. J. COMBIN. 67 (2) (2017), 243–258 244 be represented is a complete graph the concept is equivalent to that of an independent set. In this paper we discuss representing complete bipartite graphs in Steiner triple systems of small order. By relating the work to configurations in Steiner triple systems we prove that the number of representations of a graph having six or fewer edges in a Steiner triple system of order m is only dependent on the value of m and so is independent of the structure of the system.