For a graph \(G\), define \(\phi(G) = \min \{\max \{d(u), d(v)\} | d(u,v) = 2\}\) if \(G\) contains two vertices at distance 2, and \(\phi(G) = \infty\) otherwise. Fan proved that every 2-connected graph on \(n\) vertices with \(\phi(G) > \frac{1}{2}n\) is hamiltonian. Short proofs of this result and a number of analogues, some known, some new, are presented. Also, it is shown that if \(G\) is 2-connected, \(\phi(G) \geq \frac{1}{2}(n-i)\) and \(G – \{v \in V(G) | d(v) \geq \frac{1}{2} (n-i)\}\) has at least three components with more than \(i\) vertices, then \(G\) is hamiltonian (\(i \geq1\)).

We state here that, for modulus \(m\) odd and less than \(2^{29}+2^{27} – 1\), no (nontrivial) perfect binary arithmetic code, correcting two errors or more, exists (this is to be taken with respect to the Garcia-Rao modular distance). In particular, in the case \(m = 2^n \pm 1\), which is most frequently studied, no such code exists for \(m < 2^{33} – 1\).

Constructions of partially balanced incomplete block designs with three and four associate classes are given. The constructions use \(\epsilon\)-designs for \(t=6\) and \(t=8\).

Let \(X\) be a finite set of order \(mn\), and assume that the points of \(X\) are arranged in an array of size \(m \times n\). The columns of the array will be called groups.
In this paper we consider a new type of group divisible designs called modified group divisible designs in which each \(\{x,y\} \subseteq X\) such that \(x\) and \(y\) are neither in the same group nor in the same row occurs \(\lambda\) times. This problem was motivated by the problem of resolvable group divisible designs with \(k = 3\), \(\lambda = 2\) [1] , and other constructions of designs.

FE. Bennett has proved that a \((v, 4, 1)\)-RPMD exists for every positive integer \(v \equiv 1 \pmod{4}\) with the possible exception of \(v = 33, 57, 93\) and \(133\). In this paper, we shall first introduce the concept of an incomplete PMD and use it to establish some construction methods for Mendelsohn designs; then we shall give the following results: (1) a \((v, 4, 1)\)-PMD exists for every positive integer \(v \equiv 0 \pmod{4}\) with the exception of \(v = 4\) and the possible exception of \(v = 8, 12\);(2) a \((v, 4, 1)\)-PMD exists if \(v = 57, 93\) or \(133\).