A Freeman-Youden rectangle (FYR) is a Graeco-Latin row-column design consisting of a balanced superimposition of two Youden squares. There are well known infinite series of FYRs of size \(q \times (2q+1)\) and \((q+1) \times (2q+1)\) where \(2q+1\) is a prime power congruent to \(3\) (modulo \(4\)). However, Preece and Cameron [9] additionally gave a single FYR of size \(7 \times 15\). This isolated example is now shown to belong to one of a set of infinite series of FYRs of size \(q \times (2q+1)\) where \(q\), but not necessarily \(2q+1\), is a prime power congruent to \(3\) (modulo \(4\)), \(q > 3\); there are associated series of FYRs of size \((q+1) \times (2q+1)\). Both the old and the new methodologies provide FYRs of sizes \(q \times (2q+1)\) and \((q+1) \times (2q+1)\) where both \(q\) and \(2q+1\) are congruent to \(3\) (modulo \(4\)), \(q > 3\); we give special attention to the smallest such size, namely \(11 \times 23\).

Let \(n_4(k,d)\) and \(d_4(n, k)\) denote the smallest value of \(n\) and the largest value of \(d\), respectively, for which there exists an \([n, k, d]\) code over the Galois field \(GF(4)\). It is known (cf. Boukliev [1] and Table B.2 in Hamada [6]) that (1) \(n_4(5, 179) =240\) or \(249\), \(n_4(5,181) = 243\) or \(244, n_4(5, 182) = 244\) or \(245, n_4(5, 185) = 248\) or \(249\) and (2) \(d_4(240,5) = 178\) or \(179\) and \(d_4(244,5) = 181\) or \(182\). The purpose of this paper is to prove that (1) \(74(5,179) = 241, n_4(5,181) = 244, n_4(5,182) = 245, n_4(5, 185) = 249\) and (2) \(d_4(240, 5) = 178\) and \(d_4(244,5) = 181\).

Let \(T_n\) denote any rooted tree with \(n\) nodes and let \(p = p(T_n)\) and \(q = q(T_n)\) denote the number of nodes at even and odd distance, respectively, from the root. We investigate the limiting distribution, expected value, and variance of the numbers \(D(T_n) = |p – q|\) when the trees \(T_n\) belong to certain simply generated families of trees.

In this paper, magic labelings of graphs are considered. These are labelings of the edges with integers such that the sum of the labels of incident edges is the same for all vertices. We particularly study positive magic labelings, where all labels are positive and different. A decomposition in terms of basis-graphs is described for such labelings. Basis-graphs are studied independently. A characterization of an algorithmic nature is given, leading to an integer linear programming problem. Some relations with other graph theoretical subjects, like vertex cycle covers, are discussed.