A recent article of G. Chang shows that an n × n partial latin square with prescribed diagonal can always be embedded in an n × n latin square except in one obvious case where it cannot be done. Chang's proof is to show that the symbols of the partial latin square can be assigned the elements of the additive abelian group Zn so that the diagonal elements of the square sum to zero. A theorem of M. Halls then shows this to be embeddable in the operation table of the group. In this paper, we show that when n is a prime one can determine exactly the number of distinct ways in which this assignment can be made. The proof uses some graph theoretic techniques.

A recursive construction for orthogonal diagonal latin squares, using group divisible designs, is presented. In consequence the numbers of orders for which the existence of such squares is in question is reduced to 72.

Recently, we have introduced the notion of stable permutations in a Latin rectangle L(r×c) of r rows and c columns. In this note, we prove that the set of all stable permutations in L (r×c) forms a distributive lattice which is Boolean if and only if c ≤ 2.

An equidistant permutation array is a ν × r array A(r, λ;ν) defined on a r-set X such that every row of A is a permutation of X and any two distinct rows agree in precisely λ common columns. Define In this paper, we show that where n = r − λ. Certain results pertaining to irreducible equidistant permutation arrays are also established.

A directed packing of pairs into quadruples is a collection of 4-subsets of a set of cardinality ν with the property that each ordered pair of elements appears at most once in a 4-subset (or block). The maximal number of blocks with this property is denoted by DD(2, 4, ν). Such a directed packing may also be thought of as a packing of transtivie tournaments into the complete directed graph on ν points. It is shown that, for all but a finite number of values of ν, DD(2, 4, ν) is maximal.

The simple twills on n harnesses can be classified according to the number of breaks that they possess. An algorithm is detailed for determining these twills and some sample listings given. A formula is derived which evaluates the total number of n-harness twills with a specified number of breaks, and hence also the total possible number of twills on n harnesses. Also the balanced twills on n harnesses are enumerated.

New amicable orthogonal designs of order eight are given and they are used to construct ne orthogonal designs of order 32.

A Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

From an integer-valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class of matroids so obtained is closed under restriction, contraction, duality, truncation and elongation, but not under direct sum. We give an excluded-minor characterization of and show that consists precisely of those transversal matroids with a presentation in which the sets in the presentation are nested. Finally, we show that on an n-set there are exactly 2n members of .

The purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.

We give a short proof that skew Room squares exist for all odd sides s exceeding 5.

Frames have been defined as a certain type of generalization of Room square. Frames have proven useful in the construction of Room squares, in particular, skew Room squares.

We generalize the definition of frame and consider the construction of Room squares and skew Room squares using these more general frames.

We are able to construct skew Room squares of three previously unknown sides, namely 93, 159, and 237. This reduces the number of unknown sides to four: 69, 87, 95 and 123. Also, using this construction, we are able to give a short proof of the existence of all skew Room squares of (odd) sides exceeding 123.

Finally, this frame construction is useful for constructing Room squares with subsquares. We can also construct Room squares “missing” subsquares of sides 3 and 5. The “missing” subsquares of sides 3 and 5 do not exist, so these incomplete Room squares cannot be completed to Room squares.

Let g(n, m) denote the maximal number of distinct rows in any (0, 1 )-matrix with n columns, rank < n, – 1, and all row sums equal to m. This paper determines g(n, m) in all cases:

In addition, it is shown that if V is a k-dimensional vector subspace of any vector space, then V contains at most 2k vectors all of whose coordinates are 0 or 1.

We are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

Warren W. Wolfe obtained necessary conditions for the existence of orthogonal designs in terms of rational matrices. In this paper it is shown that these necessary conditions can be obtained in terms of integral matrices. In the integral form, Wolfe's theory is more useful in the construction of orthogonal designs.

We characterize all finite linear spaces with p ≤ n2 points where n ≥ 8 for p ≠ n2 − 1 and n ≥ 23 for p = n2−1, and the line range is {n−1, n, n+1}. All such linear spaces are shown to be embeddable in finite projective planes of order a function of n. We also describe the exceptional linear spaces arising from p < n2−1 and n ≥ 4.

In this paper we obtain matroid extensions of two important results in graph theory, namely the 4-colour theorem of Appel and Haken [1] and the 8-flow theorem of Jaeger [4]. As a corollary we prove that any bridgeless graph with no subgraph contractible to K3,3 has a nowhere zero 4-flow. These results depend heavily on a remarkable theory of splitters developed recently by Seymour [8], [9].

A square matrix A is transposable if P(RA) = (RA)T for some permutation matrices p and R, and symmetrizable if (SA)T = SA for some permutation matrix S. In this paper we find necessary and sufficient conditions on a permutation matrix P so that A is always symmetrizable if P(RA) = (RA)T for some permutation matrix R.

A Bhaskar Rao design is obtained from the incidence matrix of a partially balanced incomplete block design with m associate classes by negating some elements of the matrix in such a way that the inner product of rows α and β is ci if α and β are ith associates. In this paper we use nested designs constructed from unions of cyclotomic classes to give Bhaskar Rao designs.