A three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) is a three-fold resolvable packing with maximum possible number of parallel classes, each containing one block of size 3 and all other blocks of size 4. This article investigates the spectra of three-fold Kirkman packing design \( \text{KPD}_3(\{4,s^*\},v) \) for \( s = 5 \) and \( 6 \), and we show that it contains all positive integers \( v \equiv s – 4 \pmod{4} \) with \( v \geq 17 \) if \( s = 5 \), and \( v \geq 26 \) if \( s = 6 \).

Let \( G \) be a \( (p,q) \)-graph in which the edges are labeled \( 1, 2, 3, \ldots, q \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If \( G \) can be labeled so that the vertex sums are distinct, mod \( p \), then \( G \) is said to be edge-graceful. If the edges of \( G \) can be labeled \( 1, 2, 3, \ldots, q \) so that the vertex sums are constant, mod \( p \), then \( G \) is said to be edge-magic. It is conjectured by Lee [9] that any connected simple \( (p,q) \)-graph with \( q(q+1) \equiv p(p-1)/2 \pmod{p} \) vertices is edge-graceful. We show that the conjecture is true for maximal outerplanar graphs. We also completely determine the edge-magic maximal outerplanar graphs.

We enumerate the self-orthogonal Latin squares of orders \(1\) through \(9\) and discuss the nature of the isomorphism classes of each order. Furthermore, we consider the possibility of enlarging sets of self-orthogonal Latin squares to produce complete sets.

A vertex-magic total labeling on a graph with \( v \) vertices and \( e \) edges is a one-to-one map taking the vertices and edges onto the integers \( 1, 2, \ldots, v+e \) with the property that the sum of the label on a vertex and the labels of its incident edges is constant, independent of the choice of vertex. We give vertex-magic total labelings for several classes of regular graphs. The paper concludes with several conjectures and open problems in the area.

In this paper, we complete the classification of optimal binary linear self-orthogonal codes up to length 25. Optimal self-orthogonal codes are also classified for parameters up to length 40 and dimension 10. The results were obtained via two independent computer searches.

In this paper we examine the classical Williamson construction for Hadamard matrices, from the point of view of a striking analogy with isomorphisms of division algebras. By interpreting the 4 Williamson array as a matrix arising from the real quaternion division algebra, we construct Williamson arrays with 8 matrices, based on the real octonion division algebra. Using a Computational Algebra formalism we perform exhaustive searches for even-order 4-Williamson matrices up to 18 and odd- and even-order 8-Williamson matrices up to 9 and partial searches for even-order 4-Williamson matrices up to 22 and odd- and even-order 8-Williamson matrices for orders 10 — 13. Using Magma, we conduct searches for inequivalent Hadamard matrices within all the sets of matrices obtained by exhaustive and partial searches. In particular, we establish constructively ten new lower bounds for the number of inequivalent, Hadamard matrices of the consecutive orders 72, 76, 80, 84, 88, 92, 96, 100, 104 and 108.

This article continues the study of a class of non-terminating expansions of sin\( (m\alpha) \) (even \(m \geq 2 \)) which in each case possesses embedded Catalan numbers. A known series form of the sine function (said to be associated with Euler) is taken here as our basic representation, the coefficient of the general term being developed analytically in an interesting fashion and shown to be dependent on the Catalan sequence in the manner expected.
The work, which has a historical backdrop to it, is discussed in the context of prior results by the author and others.