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 \(G\).

We complete the construction of all the simple graphs with at most \(26\) vertices and transitive automorphism group. The transitive graphs with up to \(19\) vertices were earlier constructed by McKay , and the transitive graphs with \(24\) vertices by Praeger and Royle . Although most of the construction was done by computer, a substantial preparation was necessary. Some of this theory may be of independent interest.

Given a graph \(G\) and nonnegative integer \(k\), a map \(\pi: V(G) \to \{1, \ldots, k\}\) is a perfect \(k\)-colouring if the subgraph induced by each colour class is perfect. The perfect chromatic number of \(G\) is the least \(k\) for which \(G\) has a perfect \(k\)-colouring; such an invariant is a measure of a graph’s imperfection. We study here the theory of perfect colourings. In particular, the existence of perfect \(k\)-chromatic graphs are shown for all \(k\), and we draw attention to the associated extremal problem. We provide extensions to C. Berge’s Strong Perfect Graph Conjecture, and prove the existence of graphs with only one perfect \(k\)-colouring (up to a permutation of colours). The type of approach taken here can be applied to studying any graph property closed under induced subgraphs.