A \((k, \lambda)\)-semiframe of type \(g^u\) is a group divisible design of type \(g^u\) \((\chi, \mathcal{G}, \mathcal{B})\), in which \(\mathcal{B}\) is written as a disjoint union \(\mathcal{B} = \mathcal{P} \cup \mathcal{Q} \) where \(\mathcal{P} \) is partitioned into partial parallel classes of \(\chi\) (with respect to some \(G \in \mathcal{G}\)) and \(\mathcal{Q} \) is partitioned into parallel classes of \(\chi\). In this paper, new constructions for these designs are provided with some series of designs with \(k = 3\). Cyclic semiframes are discussed. Finally, an application of semiframes is also mentioned.

A solution of Dudeney’s round table problem is given when \(n\) is as follows:

1. \(n = pq + 1\), where \(p\) and \(q\) are odd primes.
2. \(n = p^e + 1\), where \(p\) is an odd prime.
3. \(n = p^e q^f + 1\), where \(p\) and \(q\) are distinct odd primes satisfying \(p \geq 5\) and \(q \geq 11\), and \(e\) and \(f\) are natural numbers.

An upper bound on the size of any collection of mutually orthogonal partial Latin squares is derived as a function of the number of compatible cells that are occupied in all squares. It is shown that the bound is strict if the number of compatible cells is small.

The quantity \(B(G) = \min \max\{|f(u)-f(v)|: (u,v) \in E(G)\}\) is called the bandwidth of a graph \(G = (V(G), E(G))\) where \(\min\) is taken over all bijections \(f: V(G) \to \{1,2,\ldots,|V(G)|\}\) called labelings. L.H. Harper presented an important inequality related to the boundary of subsets \(S \subseteq V(G)\). This paper gives a refinement of Harper’s inequality which will be more powerful in determining bandwidths for several classes of graphs.

In this paper we consider the problem of constructing magic rectangles of size \(m \times n\) where \(m\) and \(n\) are nonprime integers. What seems to be two new methods of constructing such rectangles are given.

The point-distinguishing chromatic index \(\chi_o(G)\) of a graph \(G\) represents the minimum number of colours in an edge colouring of \(G\) such that each vertex of \(G\) is distinguished by the set of colours of its incident edges. It is known that \(\chi_o(K_{n,n})\) is a non-decreasing function of \(n\) with jumps of value \(1\). We prove that \(\chi_o(K_{46,46}) = 7\) and \(\chi_o(K_{47,47}) = 8\).

There have been many results concerning claw-free graphs and hamiltonicity. Recently, Jackson and Wormald have obtained more general results on walks in claw-free graphs. In this paper, we consider the family of almost claw-free graphs that contains the previous one, and give some results on walks, especially on shortest covering walks visiting only once some given vertices.

A \(t\)-(n, k, \(\lambda\)) covering design consists of a collection of \(k\)-element subsets (blocks) of an \(n\)-element set \(\chi\) such that each \(t\)-element subset of \(\chi\) occurs in at least \(\lambda\) blocks. We use probabilistic techniques to obtain a general upper bound for the minimum size of such designs, extending a result of Erdős and Spencer [4].

In this paper, difference sets in groups containing subgroups of index \(2\) are considered, especially groups of order \(2m\) where \(m\) is odd. The author shows that the only difference sets in groups of order \(2p^\alpha\) are trivial. The same conclusion is true for some special parameters.