We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning. If the two squares are mutual transposes, we say that we have SOLS (self-orthogonal Latin squares) with holes. It is shown that a pair of SOLS with \(n\) holes of size \(h \geq 2\) exist if and only if \(n \geq 4\) and it is also shown that a pair of SOLS with \(n\) holes of size \(2\) and one hole of size \(3\) exist for all \(n \geq 4, n \neq 13, 15\).

As an application, we prove a result concerning intersection numbers of transversal designs with four groups.

Let \(V\) be a finite set of order \(v\). A \((v, \kappa, \lambda)\) covering design of index \(\lambda\) and block size \(\kappa\) is a collection of \(\kappa\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks, \(\alpha(v, \kappa, \lambda)\), in a covering design. It is well known that
\(\alpha(v, \kappa, \lambda) \geq \lceil \frac{v}{\kappa}\lceil\frac{v-1}{\kappa -1}\lambda\rceil\rceil = \phi(v, \kappa, \lambda)\)
where \(\lceil x \rceil\) is the smallest integer satisfying \(x \leq \lceil x \rceil\). It is shown here that
\(\alpha(v, 5, 6) = \phi (v, 5, 6)\) for all positive integers \(v \geq 5\), with the possible exception of \(v = 18\).

In an edge-colored graph, a cycle is said to be alternating, if the successive edges in it differ in color. In this work, we consider the problem of finding alternating cycles through \(p\) fixed vertices in \(k\)-edge-colored graphs, \(k \geq 2\). We first prove that this problem is NP-Hard even for \(p = 2\) and \(k = 2\). Next, we prove efficient algorithms for \(p = 1\) and \(k\) non-fixed, and also for \(p = 2\) and \(k = 2\), when we restrict ourselves to the case of \(k\)-edge-colored complete graphs.

It is shown that the obvious necessary condition for the existence of a \(\text{B}(8,7; v)\) is sufficient, with the possible exception of \(v \in \{48, 56, 96, 448\}\).

We prove that for any tree \(T\) of maximum degree three, there exists a subset \(S\) of \(E(T)\) with \(|S| = O(\log n)\) and a two-coloring of the edges of the forest \(T \setminus S\) such that the two monochromatic forests are isomorphic, where \(n\) is the number of vertices of \(T\) of degree three.

We construct new simple \(3-(17,5,3), 3-(19,9,56), 3-(19,9,140)\), and \(3-(19,9,224)\) designs by combining disjoint designs.

An \(\text{NB}[k, \lambda; v]\) is a \(\text{B}[b, \lambda; v]\) which has no repeated blocks. In this paper we prove that there exists an indecomposable \(\text{NB}[3,5; v]\) for \(v \geq 7\) and \(v \equiv 1 \text{ or } 3 \pmod{6}\), with the exception of \(v = 7\) and \(9\), and the possible exception of \(v = 13, 15\).

We propose several invariants for cycle systems and \(2\)-factorizations of complete graphs, and enumerate the \(4\)- and \(6\)-cycle systems of \(K_g\).

Let \(G\) be a simple connected graph on \(2n\) vertices with a perfect matching. \(G\) is \(k\)-\({extendable}\) if for any set \(M\) of \(k\) independent edges, there exists a perfect matching in \(G\) containing all the edges of \(M\). \(G\) is \({minimally \; k-extendable}\) if \(G\) is \(k\)-extendable but \(G – uv\) is not \(k\)-extendable for every pair of adjacent vertices \(u\) and \(v\) of \(G\). The problem that arises is that of characterizing \(k\)-extendable and minimally \(k\)-extendable graphs. The first of these problems has been considered by several authors whilst the latter has only been recently studied. In a recent paper, we established several properties of minimally \(k\)-extendable graphs as well as a complete characterization of minimally \((n – 1)\)-extendable graphs on \(2n\) vertices. In this paper, we focus on characterizing minimally \((n – 2)\)-extendable graphs. A complete characterization of \((n – 2)\)-extendable and minimally \((n – 2)\)-extendable graphs on \(2n\) vertices is established.