A set \(X\) of vertices of a graph is said to be dependent if \(X\) is not an independent set. For the graph \(G\), let \(P_k(G)\) denote the set of dependent sets of cardinality \(k\).

In this paper, we show that if \(G\) is a connected graph on \(2n\) vertices where \(n \geq 3\), then \(|P_n(G)| \geq |P_{n+1}(G)|\). This study is motivated by a conjecture of Lih.

The shares in a \((k,n)\) Shamir threshold scheme consist of \(n\) points on some polynomial of degree at most \(k-1\). If one or more of the shares are faulty, then the secret may not be reconstructed correctly. Supposing that at most \(\epsilon\) of the \(n\) shares are faulty, we show how a suitably chosen covering design can be used to compute the correct secret. We review known results on coverings of the desired type, and give some new constructions. We also consider a randomized algorithm for the same problem, and compare it with the deterministic algorithm obtained by using a particular class of coverings.

A finite ordered set is upper levellable iff it has a diagram in which, for each element, all upper covers of the element are on the same horizontal level. In this note, we give a method for computing a canonical upper levelling, should one exist.

An \(n_3\)-configuration in the real projective plane is a configuration consisting of \(n\) points and \(n\) lines such that every point is on three lines and every line contains three points. Determining sets are used to construct drawings of arbitrary \(n_3\)-configurations in the plane, such that one line is represented as a circle. It is proved that the required determining set always exists, and that such a drawing is always possible. This is applied to the problem of deciding when a particular configuration is coordinatizable.

For a given graph \(G\), we fix \(s\), and partition the vertex set into \(s\) classes, so that any given class contains few edges. The result gives a partition \((U_1, U_2, \ldots, U_s)\), where \(e(U_i) \leq \frac{e(G)}{s^2} + 4s\sqrt{e(G)}\) for each \(1 \leq i \leq s\). The error term is compared to previous results for \(s = 2^P\) \({[6]}\), and to a result by Bollobás and Scott \({[1]}\).

We associate codes with \(C(n,n,1)\) designs. The perfect \(C(n,n,1)\) designs obtained from perfect one-factorizations of \(K_n\) yield codes of dimension \(n-2\) over \(\mathbb{F}_2\) and \(n-1\) over \(\mathbb{F}_p\), for \(p\neq 2\). We also demonstrate a method of obtaining another \(C(n,n,1)\) design from a pair of isomorphic perfect \(C(n,n,1)\) designs and determine the dimensions of the resulting codes.

In a previous work “Skolem labelled graphs” \({[4]}\) we defined the Skolem labelling of graphs, here we prove that the necessary conditions are sufficient for a Skolem or minimum hooked Skolem labelling of all \(k\)-windmills. A \(k\)-windmill is a tree with \(k\) leaves each lying on an edge-disjoint path of length \(m\) to the centre. These paths are called the vanes.

Let \(v\), \(k\),\(\lambda\) and \(n\) be positive integers. \((x_1, x_2, \ldots, x_k)\) is defined to be \(\{(x_1, x_2), (x_2, x_3), \ldots, (x_k-1, x_k), (x_k, x_1)\}\), and is called a cyclically ordered \(k\)-subset of \(\{x_1, x_2, \ldots, x_1\}\). An incomplete perfect Mendelsohn design, denoted by \((v, n, 4, \lambda)\)-IPMD, is a triple \((X, Y, \mathcal{B})\), where \(X\) is a \(v\)-set (of points), \(Y\) is an \(n\)-subset of \(X\), and \(\mathcal{B}\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks) such that every ordered pair \((a, b) \in X \times X \setminus Y \times Y\) appears \(t\)-apart in exactly \(\lambda\) blocks of \(\mathcal{B}\) and no ordered pair \((x, y) \in Y \times Y\) appears in any block of \(\mathcal{B}\) for any \(t\), where \(1 \leq t \leq (k – 1)\). In this paper, the necessary condition for the existence of a \((v, n, 4, \lambda)\)-IPMD for even \(\lambda\), namely \(v \geq (3n + 1)\), is shown to be sufficient.