Steiner triple systems satisfying the 4-vertex condition

Abstract

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled. We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.

Strongly regular graphs with the $$7$$ 7 -vertex condition

“…While the cases m = 9 and m = 13 can be easily settled, the case m = 25 remained open for four decades. Following the advice of M. Klin, P. Kaski et al obtained the negative answer (see [26]), essentially relying on the use of computer and some extra clever ad hoc tricks (the complete enumeration of all ST S(25) looks hopeless at the moment).…”

Strongly regular graphs satisfying the 4-vertex condition

“…He found that either the system is a projective space PG(m, 2) or v is one of 9, 13, 25. In [25] the cases 13 and 25 are ruled out, so that the only other example is the affine plane AG (2,3). The examples are rank 3.…”

On highly regular strongly regular graphs

“…As non-rank 3 srgs with the t-vertex condition for t > 3 appear to be very rare, there has been an ongoing research effort to discover new examples and to understand their nature (cf. [18,19,23,24,29,31]).…”