Let \(V\) be a finite set of order \(\nu\). A \((\nu,\kappa,\lambda)\) packing design of index \(\lambda\) and block size \(\kappa\) is a collection of \(\kappa\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at most \(\lambda\) blocks. The packing problem is to determine the maximum number of blocks, \(\sigma(\nu,\kappa,\lambda)\), in a packing design. It is well known that \(\sigma(\nu,\kappa,\lambda) < \left[ \frac{\nu}{\kappa}[\frac{(\nu-1)}{\kappa(\kappa-1)}] \right] = \psi(\nu,\kappa,\lambda)\), where \([x]\) is the largest integer satisfying \(x \ge [x]\). It is shown here that if \(v \equiv 2 \pmod{4}\) and \(\nu \geq 6\) then \(\sigma(\nu,5,3) = \psi(\nu,5,3)\) with the possible exception of \(v = 38\).

In this paper we obtain some new relations on generalized exponents of primitive matrices. Hence the multiexponent of primitive tournament matrices are evaluated.

The ranking and unranking problem of a Gray code \(C(n,k)\) for compositions of \(n\) into \(k\) parts is solved. This means that rules have been derived by which one can calculate in a non-recursive way the index of a given codeword, and vice versa, determine the codeword with a given index. A number system in terms of binomial coefficients is presented to formulate these rules.

In the definition of local connectivity, the neighbourhood of a vertex consists of the induced subgraph of all vertices at distance one from the vertex. In {[2]}, we introduced the concept of distance-\(n\) connectivity in which the distance-\(n\) neighbourhood of a vertex consists of the induced subgraph of all vertices at distance less or equal to \(n\) from that vertex. In this paper we present Menger-type results for graphs whose distance-\(n\) neighbourhoods are all \(k\)-connected, \(n \geq 1\).

A partially ordered set \(P\) is called a circle order if one can assign to each element \(a \in P\) a circular disk in the plane \({C_a}\), so that \(a < b\) iff \(C_a \subset C_b\). It is known that the dual of every finite circle order is a circle order. We show that this is false for infinite circle orders.

In [Discrete Math. 46 (1983) 191 – 198], the concept of inclusive edge connectivity was introduced and discussed. Given a vertex \(v \in V(G)\), the inclusive edge connectivity of \(v\), denoted by \(\lambda_i(v,G)\), is the minimum number of edges whose deletion results in a subgraph of \(G\) in which \(v\) is a cut-vertex. Define

\[\lambda_i(v,G) = \min\{\lambda_i(v,G) : v \in V(G), \text{ and } d_G(v) \geq 2\}\]

to be the inclusive edge connectivity of \(G\). Extremal problems on \(\lambda_i(G)\) are studied in this paper.

If the binding number of a graph \(G\) is more than \(1 + \frac{a-1}{b}\), does \(G\) have an \([a,b)\)-factor? The answers to this question for the case of either \(a = b\) or \(a \leq 3\) can be found in [1], [2], [4], and [7]. Here we give some more answers for \(4 \leq a \leq b\).

The design of de Bruijn sequences is equivalent to finding spanning trees in certain graphs. We give an algorithm which finds spanning
trees in these graphs using the universal circuit matrix defined in \([9]\).

In this note, we obtain nonexistence results for \((m,2,m-1,\frac{m-2}{2})\) relative difference sets. In particular, we obtain further restriction on the parameters of splitting \((m,2,m-1,\frac{m-2}{2})\) relative difference set under certain condition.

We define a new embedding invariant, namely \(n\)-polyhedrality, and we propose a program of research in which the objective is to enumerate the \(n\)-polyhedral embeddings of a given graph for various values of \(n\). We begin the program for the cartesian products of cycles by showing that \(C_3 \times C_n\) has exactly one \(3\)-polyhedral embedding.