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We prove that an almost $$2$$ -perfect $$6$$ -cycle system of order $$n$$ exists if and only if $$n \equiv 1$$ or $$9\ (mod\ 12)$$ , and that an almost $$2$$ -perfect maximum packing with $$6$$ -cycles of order $$n$$ exists for all $$n \ge 6$$ . [ABSTRACT FROM AUTHOR]

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included. [ABSTRACT FROM AUTHOR]

A c-hybrid triple system of order v is a decomposition of the complete v-vertex digraph into c cyclic tournaments of order 3 and $$\frac{{v(v - 1)}}{3} - c$$ transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems ( c = 0) and Mendelsohn triple systems ( c = v(v − 1)/3); omitting directions yields an underlying twofold triple system. The spectrum of v and c for which a c-hybrid triple system of order v exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order $$v\left( {having b_v = \frac{{v(v - 1)}}{3}blocks} \right)$$ underlies a c-hybrid triple system with $$c \geqslant \frac{{2b_v }}{3}$$ . Examples are constructed for all sufficiently large v, for which this maximum is at most $$\left( {\frac{7}{{10}} + \varepsilon } \right)b_v $$ . The lower bound here is proved by establishing bounds on F( n, r), the size of minimum cardinality vertex feedback sets in n-vertex i-connected cubic multigraphs having r repeated edges. We establish that $$F_0 (n,r) \leqslant \frac{n}{2},$$ , $$F_1 (n,r) \leqslant \frac{{3n}}{8} + \frac{r}{4} + \frac{1}{2}, and F_2 (n,r) \leqslant \frac{{(n + r)}}{3}for n > 8$$ . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem. [ABSTRACT FROM AUTHOR]

Every graph having vertex degrees zero and two satisfying the basic necessary conditions is the leave of a maximal partial triple system, with one exception ( C ⋃ C). The proof technique is direct, using the method of differences. [ABSTRACT FROM AUTHOR]