The minimum cardinality of a pairwise balanced design on nineteen points is determined; a minimal design is exhibited containing \(13\) triples and \(22\) quadruples.

It is shown that the collection of all the \(\dbinom{10}{3}\) triples chosen from a set of ten points can be partitioned into ten mutually disjoint \(2-(9,3,1)\) designs in precisely \(77\) non-isomorphic ways.

A \((3,k,n,e)\) Ramsey graph is a triangle-free graph on \(n\) vertices with \(e\) edges and no independent set of size \(k\). Similarly, a \((3,k)\)-, \((3,k,n)\)-, or \((3,k,n,e)\)-graph is a \((3,k,n,e)\) Ramsey graph for some \(n\) and \(e\). In the first part of the paper, we derive an explicit formula for the minimum number of edges in any \((3,k,n)\)-graph for \(n\leq3(k-1)\), i.e., a partial formula for the function \(e(3,k,n)\) investigated in \([3,5,7]\). We prove some general properties of minimum \((3,k,n)\)-graphs with \(e(3,k,n)\) edges and present a construction of minimum \((3,k+1,3k-1,5k-5)\)-graphs for \(k\geq2\) and minimum \((3,k+1,3k,5k)\)-graphs for \(k\geq4\). In the second part of the paper, we describe a catalogue of small Ramsey graphs: all \((3,k)\)-graphs for \(k\leq6\) and some \((3,7)\)-graphs, including all \(191 (3,7,22)\)-graphs, produced by a computer. We present for \(k\leq7\) all minimum \((3,k,n)\)-graphs and all \(10\) maximum \((3,7,22)\)-graphs with \(66\) edges.

The cost of a sorting algorithm is the number of primitive operations used in a worst-case execution of the algorithm. In the standard model, the primitive operation is a binary comparison, which sorts a pair of keys. Cost measures based on other primitive operations are considered. A general lower bound for the cost of a sorting algorithm is given, which is valid for a wide class of possible primitives. For several special primitive operations, sorting algorithms are given. The primitive operations studied are: sorting \(k\) keys, finding the largest among \(k\) keys, and merging lists of lengths \(i,j\).

Hartman and Rosa have shown that the complete graph \(K_{2n}\) has an acyclic one-factorization if and only if \(n\) is not a power of \(2\) exceeding \(2\). Here, we consider the following problem: for which \(n > 0\) and \(0 < k < \frac{n}{2}\) does the complete graph \(K_n\) admit a cyclic decomposition into matchings of size \(k\)? We give a complete solution to this problem and apply it to obtain a new class of perfect coverings.

The integrity of a graph \(G\), denoted \(I(G)\), is defined by \(I(G) := \min_{S \subset V(G)} \{|S| + m(G – S)\}\), where \(m(G – S)\) denotes the maximum order of a component of \(G – S\). In this paper, we explore the integrity of various combinations of graphs in terms of the integrity and other graphical parameters of the constituent graphs. Specifically, explicit formulae and/or bounds are derived for the integrity of the join, union, cartesian and lexicographic products of two graphs. Also, some results on the integrity of powers of graphs are given.