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.

We prove that the smallest covering code of length \(8\) and covering radius \(2\) has exactly \(12\) words. The proof is based on partial classification of even weight codewords, followed by a search for small sets of odd codewords covering the part of the space that has not been covered by the even subcode.

Alon and Yuster {[4]} have proven that if a fixed graph \(K\) on \(g\) vertices is \((h+1)\)-colorable, then any graph \(G\) with \(n\) vertices and minimum degree at least \(\frac{h}{h+1}n\) contains at least \((1-\epsilon)\frac{n}{g})\) vertex disjoint copies of \(K\), provided \(n>N(\epsilon)\). It is shown here that the required minimum degree of \(G\) for this result to follow is closer to \(\frac{h-1}{h }n\), provided \(K\) has a proper \((h+1)\)-coloring in which some of the colors occur rarely. A conjecture regarding the best possible result of this type is suggested.

Let \(G\) be a finite group with a normal subgroup \(H\). We prove that if there exist a \((h, r;\lambda, H)\) difference matrix and a \((g/h, r;1, G/H)\) difference matrix, then there exists a \((g, r;\lambda, G)\) difference matrix. This shows in particular that if there exist \(r\) mutually orthogonal orthomorphisms of \(H\) and \(r\) mutually orthogonal orthomorphisms of \(G/H\), then there exist \(r\) mutually orthogonal orthomorphisms of \(G\). We also show that a dihedral group of order \(16\) admits at least \(3\) mutually orthogonal orthomorphisms.

Let \(k\) and \(b\) be integers and \(k > 1\). A set \(S\) of integers is called \((k, b)\) linear-free (or \((k, b)\)-LF for short) if \(2 \in S\) implies \(kx + b \notin S\). Let \(F(n, k, b) = \max\{|A|: A \text{ is } (k, 0)\text{-LF and } A \subseteq [1, n]\}\), where \([1, n]\) denotes all integers between \(1\) and \(n\). A subset \(A\) of \([1, n]\) with \(|A| = F(n, k, b)\) is called a maximal \((k, b)\)-LF subset of \([1, n]\). In this paper, a recurrence relation for \(F(n, k, b)\) is obtained and a method to construct a maximal \((k, b)\)-LF subset of \([1, n]\) is given.