In this paper we construct pairwise balanced designs (PBDs) having block sizes which are prime powers congruent to \(1\) modulo \(5\) together with \(6\). Such a PBD contains \(n = 5r + 1\) points, for some positive integer \(r\). We show that this condition is sufficient for \(n \geq 1201\), with at most \(74\) possible exceptions below this value. As an application, we prove that there exists an almost resolvable BIB design with \(n\) points and block size five whenever \(n \geq 991\), with at most \(26\) possible exceptions below this value.

A Nuclear Design \(ND(v; k, \lambda)\) is a collection \( {B}\) of \(k\)-subsets of a \(v\)-set \(V\), where \( {B} = \mathcal{P}\cap {C} \), where \((V, \mathcal{P})\) is a maximum packing \((PD(v; k,\lambda))\) and \((V, \mathcal{C})\) is a minimum covering \((CD(v; k,\lambda))\) with \(|{B}|\) as large as possible. We construct \(ND(v; 3, 1)\)’s for all \(v\) and \(\lambda\). Along the way we prove that for every leave (excess) possible for \(k = 3\), all \(v,\lambda\), there is a maximum packing (minimum covering) achieving this leave (excess).

A graph \(G\) is defined to be balanced if its average degree is at least as large as the average degree of any of its subgraphs. We obtain a characterization of all balanced graphs with minimum degree one. We prove that maximal \(Q\) graphs are strictly balanced for several hereditary properties \(Q\). We also prove that a graph \(G\) is balanced if and only if its subdivision graph \(S(G)\) is balanced.

In “On the exact minimal (1, 4)-cover of twelve points” (\textit{Ars Combinatoria} 27, 3–18, 1989), Sane proved that if \(E\) is an exact minimal (1, 5)-cover of nineteen points, then \(E\) has 282, 287, 292, or 297 blocks. Here we rule out the first possibility.

It is shown that a \(4\)-critical planar graph must contain a cycle of length \(4\) or \(5\) or a face of size \(k\), where \(6 \leq k \leq 11\).

We give a construction of a row-complete Latin square, which cannot be made column-complete by a suitable permutation of its rows, for every even order greater than \(8\).