Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc 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 ofv andc for which ac-hybrid triple system of orderv 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 ac-hybrid triple system with$$c \geqslant \frac{{2b_v }}{3}$$. Examples are constructed for all sufficiently largev, for which this maximum is at most$$\left( {\frac{7}{{10}} + \varepsilon } \right)b_v $$. The lower bound here is proved by establishing bounds onFi(n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr 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.