Symmetric balanced squares for different sizes of array and for different numbers of treatments have been constructed. An algorithm, easily implementable on computers, has been developed for construction of such squares whenever the parameters satisfy the necessary conditions for existence of the square. The method of construction employs \(1\)-factorizations of a complete graph or near \(1\)-factorizations of a complete graph, depending on whether the size of the array is even or odd, respectively. For odd sized squares the method provides a solution directly based on the near \(1\)-factorization. In the case of the squares being of even size, we use Hall’s matching theorem along with a \(1\)-factorization if \([\frac{n^2}{v}]\) is even, otherwise, Hall’s matching theorem together with Fulkerson’s~\([4]\) theorem, on the existence of a feasible flow in a network with bounds on flow leaving the sources and entering the sinks, lead to the required solution.

This paper presents a probabilistic polynomial-time reduction of the discrete logarithm problem in the general linear group \(\mathrm{GL}(n, \mathbb{F})\) to the discrete logarithm problem in some small extension fields of \(\mathbb{F}_p\).

A distance two labelling (or coloring) is a vertex labelling with constraints on vertices within distance two, while the regular vertex coloring only has constraints on adjacent vertices (i.e. distance one). In this article, we consider three different types of distance two labellings. For each type, the minimum span, which is the minimum range of colors used, will be explored. Upper and lower bounds are obtained. Graphs that attain those bounds will be demonstrated. The relations among the minimum spans of these three types are studied.

Arcs and linear maximum distance separable \((M.D.S.)\) codes are equivalent objects~\([25]\). Hence, all results on arcs can be expressed in terms of linear M.D.S. codes and conversely. The list of all complete \(k\)-arcs in \(\mathrm{PG}(2,q)\) has been previously determined for \(q \leq 16\). In this paper, (i) all values of \(k\) for which there exists a complete \(k\)-arc in \(\mathrm{PG}(2,q)\), with \(17 \leq q \leq 23\), are determined; (ii) a complete \(k\)-arc for each such possible \(k\) is exhibited.

An \((r,s; m,n)\)-de Bruijn array is a periodic \(r \times s\) binary array in which each of the different \(m \times n\) matrices appears exactly once. C.T. Fan, S.M. Fan, S.L. Ma and M.K. Siu established a method to obtain either an \((r,2^n;m+1,n)\)-array or a \((2r,2^{n-1};m+1,n)\)-array from an \((r,s; m, n)\)-array. A class of square arrays are constructed by their method. In this paper, decoding algorithms for such arrays are described.

An \(H\)-transformation on a simple \(3\)-connected cubic planar graph \(G\) is the dual operation of flip flop on the triangulation \(G^*\) of the plane, where \(G^*\) denotes the dual graph of \(G\). We determine the seven \(3\)-connected cubic planar graphs whose \(H\)-transformations are uniquely determined up to isomorphism.

Conditions are given for decomposing \(K_{m,n}\) into edge-disjoint copies of a bipartite graph \(G\) by translating its vertices in the bipartition of the vertices of \(K_{m,n}\). A construction of the bipartite adjacency matrix of the \(d\)-cube \(Q_d\) is given leading to a convenient \(\alpha\)-valuation and a proof that \(K_{d2^{d-2},d2^{d-1}}\) can be decomposed into copies of \(Q_d\) for \(d > 1\).

Let \(G\) be a connected graph of order \(n\) and let \(k\) be a positive integer with \(kn\) even and \(n \geq 8k^2 + 12k + 6\). We show that if \(\delta(G) \geq k\) and \(\max\{d(u), d(v)\} \geq n/2\) for each pair of vertices \(u,v\) at distance two, then \(G\) has a \(k\)-factor. Thereby a conjecture of Nishimura is answered in the affirmative.