We verify that \(6\) more of the tum squares of order \(10\) cannot be completed to a triple of mutually orthogonal Latin squares of order \(10\). We find a pair of orthogonal Latin squares of order \(10\) with \(6\) common transversals, \(5\) of which have only a single intersection, and a pair with \(7\) common transversals.

We give a complete solution to the existence problem for subdesigns in complementary \(\mathrm{P}_3\)-decompositions, where \(\mathrm{P}_3\) denotes the path of length three. As a corollary we obtain the spectrum for incomplete designs with block size four and \(\lambda = 2\), having one hole.

In this paper, we give some recursive constructions for large sets of disjoint group-divisible designs with block size \(3\). In particular, we construct new infinite classes of large sets for designs having group-size two. These large sets have applications in cryptography to the construction of perfect threshold schemes.

A decomposition into non-isomorphic matchings, or \(DINIM\) for short, is a partition of the edges of a graph \(G\) into matchings of different sizes. As a special case of our results, we prove that if a graph \(G\) has at least \((2\chi’ – 2)\chi’ + 1\) edges, where \(\chi’ = \chi'(G)\) is the chromatic index of \(G\), then \(G\) has a \(DINIM\). In particular, the \(n\)-dimensional cube, \(Q_n\), \(n \geq 4\), has a \(DINIM\). These results confirm two conjectures which appeared in Chinn and Richter [3].

We present a permutation group whose orbits classify isomorphism of covering projections of the complete graph with \(4\) vertices. Two structure theorems concerning this group are proved.

Constructions have been completed which improve the lower bounds for \(R(4,6)\), \(R(5,6)\) and \(R(3,12)\).

Let \(G\) be a graph with minimum degree \(\delta\). For each \(i = 1, 2, \ldots, \delta \), let \(a_i(G)\) (resp. \(\alpha^*_i(G)\)) denote the smallest integer \(k\) such that \(G\) has an \([i, k]\)-factor (resp. a connected \([i, k]\)-factor). Denote by \(G_n\) a complete \(n\)-partite graph. In this paper, we determine the value of \(\alpha_t(G_n)\), and show that \(0 \leq \alpha^*_1(G_n) – \alpha(G_n) \leq 1\) and \(\alpha^*_i(G_n) = a_i(G_n)\) for each \(i = 2, 3, \ldots, \delta\).

An oriented (or ordered) triple means either a Mendelsohn or a transitive triple. An oriented (or ordered) triple system of order \(v\), briefly OTS(\(v\)), is a pair \((V, B)\), where \(V\) is a \(v\)-set and \(B\) is a collection of oriented triples of elements of \(V\), such that every ordered pair of distinct elements of \(V\) belongs to exactly one member of \(B\). It is known that an OTS(\(v\)) exists if and only if \(v \equiv 0, 1 \pmod{3}\). An OTS(\(v\)) is cyclic if it admits an automorphism consisting of a single cycle of length \(v\); an OTS(\(v\)) is rotational if it admits an automorphism consisting of a single fixed point and one cycle of length \(v-1\). In this note we give some constructions of OTS(\(v\))’s which allow to determine the spectrum of cyclic and of rotational OTS(\(v\))’s.

It is shown that the maximal number of pairwise edge disjoint trees of order seven in the complete graph \(K_n\), and the minimum number of trees of order seven, whose union is \(K_n\), are \(\left\lfloor\frac{n(n-1)}{12}\right\rfloor\) and \(\left\lceil\frac{n(n-1)}{12}\right\rceil,n\geq 11\), respectively. (\(\lfloor x\rfloor\) denotes the largest integer not exceeding \(x\) and \(\lceil x\rceil\) the least integer not less than \(x\)).