A \((v,k,t)\) trade \(T = T_1 – T_2\) of volume \(m\) consists of two disjoint collections \(T_1\) and \(T_2\), each containing \(m\) blocks (\(k\)-subsets) such that every \(t\)-subset is contained in the same number of blocks in \(T_1\) and \(T_2\). If each \(t\)-subset occurs at most once in \(T_1\), then \(T\) is called a Steiner \((k,t)\) trade. In this paper, the spectrum (that is, the set of allowable volumes) of Steiner trades is discussed, with particular reference to the case \(t = 2\). It is shown that the volume of a Steiner \((k, 2)\) trade is at least \(2k – 2\) and cannot equal \(2k – 1\). We show how to construct a Steiner \((k, 2)\) trade of volume \(m\) when \(m \geq 3k – 3\), or \(m\) is even and \(2k – 2 \leq m \leq 3k – 4\). For \(k = 5\) or \(6\), the non-existence of Steiner \((k,2)\) trades of volume \(2k + 1\) is demonstrated, and for \(k = 7\), we exhibit a Steiner \((k,2)\) trade of volume \(2k + 1\). In addition, the structure of Steiner \((k,2)\) trades of volumes \(2k – 2\) and \(2k\) (\(k \neq 3,4\)) is shown to be unique. A generalisation of our constructions to trades with blocks based on arbitrary simple graphs is also presented.

This paper characterizes a particular scheme of partially filled Latin squares and when they can be completed to full Latin squares. In particular, given an \(n \times n\) array with the first \(s\) rows and the first \(d\) cells of row \(s+1\) filled with \(n\) distinct symbols in such a way that no symbol occurs more than once in any row or column, necessary and sufficient conditions are found for when this array can be completed to a full Latin square.

We give counterexamples for two theorems given for the integrity of prisms and ladders in [2] (Theorem 2.17 and Theorem 2.18 in [1]). We also compute the integrity of several special graphs.

We apply a lattice point counting method due to Blass and Sagan [2] to compute the characteristic polynomials for the subspace arrangements interpolated between the Coxeter hyperplane arrangements. Our proofs provide combinatorial interpretations for the characteristic polynomials of such subspace arrangements. In the process of doing this, we explore some interesting properties of these polynomials.

A graph of a puzzle is obtained by associating each possible position with a vertex and by inserting edges between vertices if and only if the corresponding positions can be obtained from each other in one move. Computational methods for finding the vertices at maximum distance \(\delta\) from a vertex associated with a goal position are presented. Solutions are given for small sliding block puzzles, and methods for obtaining upper and lower bounds on \(\delta\) for large puzzles are considered. Old results are surveyed, and a new upper bound for the 24-puzzle is obtained: \(\delta \leq 210\).

The total domination number \(\gamma_t(G)\) of graph \(G = (V, E)\) is the cardinality of a smallest subset \(S\) of \(V\) such that every vertex of \(V\) has a neighbor in \(S\). It is known that, if \(G\) is a connected graph with \(n\) vertices, \(\gamma_t(G) \leq \left\lfloor{2n}/{3}\right\rfloor\). Graphs achieving this bound are characterized.

In this paper we define the imbalance of equi-replicate incomplete block designs. We prove that the imbalance measure of an equi-replicate incomplete block design has a lower bound, and this bound is attained if and only if the design is a 2-concurrence design. This result allows one to formulate the construction of 2-concurrence designs as an optimization problem.

Until quite recently, very few weakly completable critical sets were known. The purpose of this note is to prove the existence of at least one Latin square of each order greater than four in which a weakly completable set exists. This is done by actual construction of such a square. Non-existence of weakly completable sets in Latin squares of orders 2, 3, and 4 is already known.

In our recent paper Necessary and sufficient conditions for some two variable orthogonal designs in order 44, Koukouvinos, Mitrouli and Seberry leave 7 cases unresolved. Using a new algorithm given in our paper A new algorithm for computer searches for orthogonal designs by the present four authors we are able to finally resolve all these cases.

This note records that the necessary conditions for the existence of two variable designs constructed using four circulant matrices are sufficient. In particular, of 484 potential cases, 404 cases have been found, 68 cases do not exist, and 12 cases cannot be constructed using four circulant matrices.