Minimum embedding of Steiner triple systems into (K4−e)-designs II

Abstract

A (K 4 − e)-design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v/3, and that when equality is met, such a minimum embedding of an STS(v) exists, except when v = 15. Equality only holds when v ≡ 15, 27 (mod 30). One natural question is: What is the smallest order w such that some STS(v) can be embedded into a (K 4 − e)-design of order v + w? We solve the problem for 7 of the 10 congruence classes modulo 30.