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.

We give the numbers of nonisomorphic \(2-(7,3,\lambda)\) block designs for \(\lambda = 6,7,8,9\). We discuss the method of generation and present statistics concerning automorphism groups and multiple blocks. The \(418\) \(2-(7, 3, 6)\) block designs together with the order of their automorphism groups are listed.