Solution of Kirkman’s schoolgirl problem

A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs

“…Then S is a Steiner triple system on 8r + 1 points. If r ≡ 1(mod 3), then 3 | 8r + 1 and according to [13] there exists a parallel class of triples. Thus for each Hadamard matrix of order 4r with r ≡ 1(mod 3) we obtain an srg with parameters (4r (8r + 1), (4r + 1)(4r − 1), 2(2r − 1)(2r + 1), r (8r − 2)).…”

An algebraic property of the totally symmetric loops associated with Kirkman-Steiner triple systems

“…While Reiss [9] has shown the sufficiency of n = 1, 3(mod 6) for the existence of a Steiner triple system of order n, Ray-Chaudhuri and Wilson [8] have proved the sufficiency of n = 3(mod 6) for the existence of a Kirkman-Steiner triple system of order n.…”

The Hamilton—Waterloo problem: The case of triangle‐factors and one Hamilton cycle

“…Kirkman in 1850) is of the existence a 2-factorization of K n where each 2-factor is the union of n 3 disjoint 3-cycles -a so-called Kirkman triple system or KTS(n) (Kirkman constructed a KTS(15)). It was shown in 1971 by Ray-Chadhuri and Wilson [11] and independently by Lu (see [10]) that a KTS(n) exists if and only n ≡ 3 (mod 6). Generalizing to higher k, a resolvable k−cycle system of order n is a 2-factorization of K n in which each 2-factor consists exclusively of k−cycles.…”