Standard doubling and tripling constructions for block designs with block size three (triple systems) employ factorizations of complete graphs and of complete bipartite graphs. In these constructions, repeated edges in a factor lead to repeated blocks in the design. Hence the construction of triple systems with a prescribed number of repeated blocks is facilitated by determining the possible structure of repeated edges in the factors of a factorization of \(\lambda K_n\) and \(\lambda K_{n,n}\). For \(\lambda =3\), a complete determination of the possible combinations of numbers of doubly and triply repeated edges in 3-factorizations of \(\lambda K_n\) has been completed for \(n \geq12\). In this paper, we solve the analogous problem for the complete bipartite graphs in the case \(\lambda=3\). The case \(\lambda=1\) is trivial, and the case \(\lambda=2\) has been previously solved by Fu.

We obtain new base sequences, that is four sequences of lengths \(m + p\), \(m + p\), \(m\), \(m\), with \(p\) odd, which have zero auto correlation function which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) auto correlation function to form longer sequences.

We give an alternate construction for \(T\)-sequences of length \((4n + 3)(2m + p)\), where \(n\) is the length of a Yang nice sequence.

These results are then used in the Goethals-Seidel or (Seberry) Wallis-Whiteman construction to determine eight possible decompositions into squares of \((4n + 3)(2m + p)\) in terms of the decomposition into squares of \(2m + 1\) when there are four suitable sequences of lengths \(m + 1\), \(m + 1\), \(m\), \(m\) and \(m\), the order of four Williamson type matrices. The new results thus obtained are tabulated giving \({OD}(4t; t, t, t, t)\) for the new orders \(t \in \{121, 135, 217, 221, 225, 231, 243, 245, 247,\)\( 253, 255, 259, 261, 265, 273,\) \(275, 279, 285, 287, 289, 295, 297, 299\}\).

The Hadamard matrix with greatest known excess for these new \(t\) is then listed.

We determine those pairs \((k,v)\), \(v = 4\cdot2^m, 5\cdot2^m\), for which there exists a pair of Steiner quadruple systems on the same \(v\)-set, such that the quadruples in one system containing a particular point are the same as those in the other system and moreover the two systems have exactly \(k\) other quadruples in common.

The point set “oval” has been considered in Steiner triple systems \((STS)\) and Steiner quadruple systems \((SQS)\) [3],[2]. There are many papers about “subsystems” in \(STS\) and \(SQS\). Generalizing and modifying the terms “oval” and “subsystem” we define the special point sets “near-oval” and “near-system” in Steiner quadruple systems. Considering some properties of these special point sets we specify how to construct \(SQS\) with near-ovals (\(S^{no}\)) and with near-systems (\(S^{ns}\)), respectively. For the same order of the starting system we obtain non-isomorphic systems \(S^{no}\) and \(S^{ns}\).

P. Paulraja recently showed that if every edge of a graph \(G\) lies in a cycle of length at most \(5\) and if \(G\) has no induced \(K_{i,s}\) as a subgraph, then \(G\) has a spanning closed trail. We use a weaker hypothesis to obtain a stronger conclusion. We also give a related sufficient condition for the existence of a closed trail in \(G\) that contains at least one end of each edge of \(G\).

Let \(G\) be a \(p\)-vertex graph which is rooted at \(x\). Informally, the rotation number \(h(G, x)\) is the smallest number of edges in a \(p\)-vertex graph \(F\) such that, for every vertex \(y\) of \(F\), there is a copy of \(G\) in \(F\) with \(x\) at \(y\). In this article, we consider rotation numbers for the generalized star graph consisting of \(k\) paths of length \(n\), all of which have a common endvertex, rooted at a vertex adjacent to the centre. In particular, if \(k = 3\) we determine the rotation numbers to within \(1, 2\) or \(3\) depending on the residue of \(n\) modulo \(6\).