An extension of a method of Hammer, Sarvate and Seberry is given. As a result, from an \( {OD}(s_1,s_2,…s_r)\) of order \(n\) and a \( w(nm, p)\) an \( {OD}(ps_1,ps_2…ps_r)\) of order \(nm(n+k)\) for each integer \(k \geq 0\) is constructed.

It has been conjectured that for any union-closed set \(A\) there exists some element which is contained in at least half the sets in \(A\).
This has recently been shown that this conjecture hold if the smallest set in \(A\) has size one or two, and also to hold if the number of sets in \(A\) is less than eleven.It is shown that the smallest set size approach is unproductive for size three. It is also shown that the conjecture holds for other conditions on the sets in \(A\), and an improved bound is derived: the conjecture holds if the number of sets in \(A\) is less than 19.

Let \(G\) be a graph. A labelling \(f: V(G) \to \{0,1\}\) is called a binary labelling of \(G\). A binary labelling \(f\) of \(G\) induces an edge labelling \(\lambda\) of \(G\) as follows:
\[\lambda(u,v) = |f(u) – f(v)|\] \quad for every edge \(uv \in E(G)\).
Let \(v_f(0)\) and \(v_f(1)\) be the number of vertices of \(G\) labelled with \(0\) and \(1\) under \(f\), and \(e_0(0)\) and \(e_1(1)\) be the number of edges labelled with \(0\) and \(1\) under \(\lambda\), respectively. Then the binary labelling \(f\) of \(G\) is said to be cordial if
\[|v_f(0) – v_f(1)| \leq 1 \quad {and} \quad |e_f(0) – e_f(1)| \leq 1.\]

A graph \(G\) is cordial if it admits a cordial labelling.

In this paper, we shall give a sufficient condition for the Cartesian product \(G \times H\) of two graphs \(G\) and \(H\) to be cordial. The Cartesian product of two cordial graphs of even sizes is then shown to be cordial. We show that the Cartesian products \(P_n \times P_n\) for all \(n \geq 2\) and \(P_n \times C_{4m}\) for all \(m\) and all odd \(n\) are cordial. The Cartesian product of two even trees of equal order such that one of them has a \(2\)-tail is shown to be cordial. We shall also prove that the composition \(C_n[K_2]\) for \(n \geq 4\) is cordial if and only if \(n \not = 2 \pmod{4}\). The cordiality of compositions involving trees, unicyclic graphs, and some other graphs are also investigated.

A \(KS_2(v;1,\lambda)\) is called indecomposable if it is not isomorphic to the direct sum of a \(KS_2(v;1,\lambda_1)\) with a \(KS_2(v ;1,\lambda_2)\) for some \(\lambda_1\) and \(\lambda_2\) which add to \(\lambda\). In this note, we show that there exists an indecomposable \(KS_2(v;1,\lambda)\) for \(v \equiv 0 \pmod{2}\), \(v \geq 4\), and \(\lambda \geq 2\).

A graph \(G\) is said to be \(m\)-neighbour-connected if the neighbour-connectivity of the graph, \(K(G) = m\). A graph \(G\) is said to be critically \(m\)-neighbour-connected if it is \(m\)-neighbour-connected and the removal of the closed neighbourhood of any one vertex yields an \((m-1)\)-neighbour-connected subgraph. In this paper, we give some upper bounds of the minimum size of the critically \(m\)-neighbour-connected graphs of any fixed order \(v\), and show that the number of edges in a minimum critically \(m\)-neighbour-connected graph with order \(v\) (a multiple of \(m\)) is \(\left\lceil\frac{1}{2}mv\right\rceil\).

We classify the finite partially ordered sets which satisfy certain homogeneity conditions. One of the conditions considered is that the automorphism group of the partially ordered set acts multiply transitively on the set of elements of the same height.

Let \(G = (V, E)\) be a graph or digraph, and let \(r\) and \(s\) be two positive integers. A subset \(U\) of \(V\) is called an \((r, s)\) dominating set if for any \(v \in V – U\), there exists \(u \in U\) such that \(d(u,v) \leq r\) and for any \(u \in U\) there exists \(u’ \in U\) (\(u’ \neq u\)) for which \(d(u’,u) \leq s\). For graphs, a \((1,1)\)-dominating set is the same as a total dominating set. The \((r, s)\)-domination number \(\delta_{r,s}(G)\) of a graph or digraph \(G\) is the cardinality of a smallest \((r,s)\)-dominating set of \(G\). Various bounds on \(\delta_{r,s}(G)\) are established including that, for an arbitrary connected graph of order \(n \geq 2\), if \(s \leq r+1\) then \(\delta_{r,s}(G) \leq \max\left(\frac{2n}{r+s+1},2\right)\), and if \(s \geq r+1\) then \(\delta_{r,s}(G) < \max\left(\frac{n}{r+1},2\right)\). Both bounds are sharp.

Adjusted orthogonal row-column designs have certain desirable properties. In this paper we give a definition of adjusted orthogonal row-column designs, summarise the known designs, give some construction methods and indicate some open problems. We briefly consider the relationship between adjusted orthogonal row-column designs and orthogonal main effects block designs.

The Pfaffian of the symbols \(a_{ij}\) with \(i<j\) has a combinatorial interpretation as the signed weight generating function of perfect matchings in the complete graph. By properly specializing the variables, this generating function reduces to the signed weight generating function for the perfect matchings in an arbitrary simple graph. We construct a weight and sign preserving bijection between two appropriately constructed spaces of permutations: permutations with even cycles and pairs of involutions without fixed points. This bijection gives a purely combinatorial proof that the determinant of a zero axial skew-symmetric matrix is equal to the square of the Pfaffian.

We construct nilpotent \(SQS\)-skeins of class \(n\), for any positive integer \(n\). These \(SQS\)-skeins are all subdirectly irreducible algebras. The nilpotent \(SQS\)-skeins of class \(n\), which are constructed in this paper, are also solvable of order \( \leq \frac{n+1}{2} \) if \(n\) is odd, and of order \(\leq 1+\frac{1}{2}n\) if \(n\) is even.