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.

Generalized Steiner Systems, \(\text{GS}(2, 3, n, g)\), are equivalent to maximum constant weight codes over an alphabet of size \(g+1\) with distance \(3\) and weight \(3\) in which each codeword has length \(n\). We construct Generalized Steiner Triple Systems, \(\text{GS}(2,3,n,g)\), when \(g=4\).

Using a computer implementation, we show that two more of the Steiner triple systems on \(15\) elements are perfect, i.e., that there are binary perfect codes of length \(15\), generating \(STS\) which have rank \(15\). This answers partially a question posed by Hergert in \({[3]}\).

We also briefly study the inverse problem of generating a perfect code from a Steiner triple system using a greedy algorithm. We obtain codes that were not previously known to be generated by such procedures.

We study \(F(n,m)\), the number of compositions of \(n\) in which repetition of parts is allowed, but exactly \(m\) distinct parts are used. We obtain explicit formulas, recurrence relations, and generating functions for \(F(n,m)\) and for auxiliary functions related to \(F\). We also consider the analogous functions for partitions.

In this paper, we prove that there exists an SCSOIDLS(\(v\)) if and only if \(v \equiv 0, 1 \pmod{4}\), other than \(v = 5\), with \(40\) possible exceptions.

By Vizing’s theorem, the chromatic index \(\chi'(G)\) of a simple graph \(G\) satisfies \(\Delta(G) \leq \chi'(G) \leq \Delta(G) + 1\); if \(\chi'(G) = \Delta(G)\), then \(G\) is class \(1\), otherwise \(G\) is class \(2\). A graph \(G\) is called critical edge-chromatic graph if \(G\) is connected, class \(2\) and \(\chi'(H) < \chi'(G)\) for all proper subgraphs \(H\) of \(G\). We give new lower bounds for the size of \(\Delta\)-critical edge-chromatic graphs, for \(\Delta \geq 9\).

A critical set in a latin square is a set of entries in a latin square which can be embedded in only one latin square. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one latin square. A critical set is strong if the embedding latin square is particularly easy to find because the remaining squares of the latin square are “forced” one at a time. A semi-strong critical set is a generalization of a strong critical set. It is proved that the size of the smallest strong or semi-strong critical set of a latin square of order \(n\) is \(\left\lfloor\frac{n^2}{4}\right\rfloor\). An example of a critical set that is not strong or semi-strong is also displayed. It is also proved that the smallest critical set of a latin square of order \(6\) is \(9\).

In this paper, it is shown that any partial extended triple system of order \(n\) and index \(\lambda \geq 2\) can be embedded in an extended triple system of order \(v\) and index \(\lambda\) for all even \(v \geq 4n + 6\). This extends results known when \(\lambda = 1\).

The edge covering number \(e(P)\) of an ordered set \(P\) is the minimum number of suborders of \(P\) of dimension at most two so that every covering edge of \(P\) is included in one of the suborders. Unlike other familiar decompositions, we can reconstruct the ordered set \(P\) from its components. In this paper, we find some familiar ordered sets of edge covering number two and then show that \(e(2^n) \to \infty\) as \(n\) gets large.