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
$\alpha (v,\kappa ,\lambda )\ge \lceil \frac{v}{\kappa }\lceil \frac{v-1}{\kappa -1}\lambda \rceil \rceil =\varphi (v,\kappa ,\lambda )$
where $\lceil x\rceil$ is the smallest integer satisfying $x\le \lceil x\rceil$. It is shown here that
$\alpha (v,5,6)=\varphi (v,5,6)$ for all positive integers $v\ge 5$, 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, $k\ge 2$. 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 $v\ge 7$ and $v\equiv 1\text{ or }3(\mathrm{mod}6)$, 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 $minimallyk-extendable$ 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.