It is known that if there are base sequences of lengths \(m+1\), \(m+1\), \(m\), \(m\) and \(y\) is a Yang number, then there are \(T\)-sequences of lengths \(y(2m + 1)\). Base sequences of lengths \(m+1\), \(m+1\), \(m\), \(m\) form \(26\), \(27\), \(28\) and some new decompositions into squares are constructed. \(T\)-sequences of lengths \(61(2m + 1)\) for some new decompositions into squares are also presented.

As a network begins losing links or nodes, eventually there is a loss in its effectiveness. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network. Many graph theoretical parameters have been used to describe the stability of communication networks, including connectivity, integrity, toughness, tenacity, and binding number. Several of these deal with two fundamental questions about the resulting graph. How many vertices can still communicate? How difficult is it to reconnect the graph? For any fixed integers \(n,p\), with \(p \geq n+1\), Harary constructed classes of graphs \(H_{n,p}\), that are \(n\)-connected with the minimum number of edges. Thus Harary graphs are examples of graphs with maximum connectivity. This property makes them useful to network designers and thus it is of interest to study the behavior of other stability parameters for the Harary graphs. In this paper, we study the tenacity of the Harary graphs.

In this note, we study a group operation on the set of all row-Latin squares of order \(n\) and, as a result, are able to provide a short disproof of the Euler conjecture for infinitely many values of \(n\). We also discuss several related conjectures.

Three general constructions for covers are given. A cover is a set of \(k\)-subsets of a \(v\)-set, \(V\), such that every \(t\)-subset of \(V\) is contained in at least one of the \(k\)-sets. These constructions use the idea of dividing the \(v\)-set into either two or three equal sized subsets. The last two constructions also use the idea of establishing a correspondence between the two equal subsets in order to facilitate the construction.

In a complete bipartite graph \(K_{s,t}\), each vertex of one vertex set is joined to each vertex of the second vertex set by exactly one edge; An Eulerian orientation of \(K_{s,t}\) assigns directions to the edges in such a way that the resulting digraph has an Eulerian dicircuit. Similarly, any Eulerian circuit of \(K_{s,t}\) ascribes directions to the edges and results in an Eulerian orientation. This paper investigates Eulerian orientations and circuits of \(K_{s,t}\). Exact solutions are presented for \(s = 2\) and \(t = 4\). Computer searches were used to obtain results for other small values of \(s\) and \(t\). These results also lead to two conjectures which deal with upper and lower bounds on the numbers of Eulerian circuits.

We survey here results and problems from the reconstruction theory of evolutionary trees, which involve enumeration and inversion.

It is proved in this paper that for any integer \(n \geq 100\), a \((v,n)\)-IODLS (incomplete orthogonal diagonal Latin squares) exists if and only if \(v \geq 3n+2\). Results for \(n=2\) are also mentioned.

In this note, we construct a \((39, \{5,6,7\}, 1)\)-PBD. Thus we have a finite generating set for the PBD-closed set \(N_5^{\infty}\) with at most three inessential elements, where \(N_5^\infty = \{1\} \cup \{v: v \geq 5\}\).

In this paper, we prove the NP-hardness of the bottleneck graph bipartition problem and study the complexity status of possible approximation algorithms for some related problems.

This paper concerns neighbour designs in which the elements of each block are arranged on the circumference of a circle. Most of the designs considered comprise a general class of balanced Ouchterlony neighbour designs, which include the balanced circuit designs of Rosa and Huang \([30]\), the neighbour designs of Rees \([29]\), and the more general neighbour designs of Hwang and Lin \([13]\). The class of Rees neighbour designs includes schemes given in 1892 by Lucas \([22]\) for round dances. Isomorphism, species, and adjugate set are defined for balanced Ouchterlony neighbour designs, and some seemingly new methods of constructing such designs are presented. A new class of quasi Rees neighbour designs is defined to cover a situation where Rees neighbour designs cannot exist but where a next best thing may be needed by experimental scientists. Even-handed quasi Rees neighbour designs and even-handed balanced Ouchterlony neighbour designs are defined too, the latter being closely related to serially balanced sequences. This paper does not provide a complete survey of known results, but aims to give the flavour of the subject and to indicate many openings for further research.