Let \(k,n\) be positive integers. Define the number \(f(k,n)\) by\\
\(f(k,n) = \min \left\{\max \left\{|S_i|: i=1,\ldots,k\right\}\right\},\)
where the minimum is taken over all \(k\)-tuples \(S_1,\ldots,S_k\) of cliques of the complete graph \(K_n\), which cover its edge set. Because there exists an \((n,m,1)\)-BIBD if and only if \(f(k,n) = m\), for \(k=\frac{n(n-1)}{m(m-1)}\), the problem of evaluating \(f(k,n)\) can also be considered as a generalization of the problem of existence of balanced incomplete block designs with \(\lambda=1\).

In the paper, the values of \(f(k,n)\) are determined for small \(k\) and some asymptotic properties of \(f\) are derived. Among others, it is shown that for all \(k\) \(\lim_{n\to\infty} \frac {f(k,n)}{n} \) exists.

A new method of construction of balanced ternary designs from PBIB designs, which yields two new designs, is given.

A dominating set \(X\) of a graph \(G\) is a k-minimal dominating set of \(G\) iff the
removal of any \(\ell \leq k\) vertices from \(X\) followed by the addition of any \(\ell \sim 1\) vertices of G
results in a set which does not dominate \(G\). The \(k\)-minimal domination number IWRC)
of \(G\) is the largest number of vertices in a k-minimal dominating set of G. The sequence
\(R:m_1 \geq m_2 \geq… \geq m_k \geq …. \geq\) n of positive integers is a domination sequence iff
there exists a graph \(G\) such that \(\Gamma_1 (G) = m_1, \Gamma_2(G) = m_2,… \Gamma_k(G) = m_k,…,\)
and \(\gamma(G) = n\), where \(\gamma(G)\) denotes the domination number of G. We give sufficient
conditions for R to be a domination sequence.

Using the definition of \(k\)-free, a known result can be re-stated as follows: If \(G\) is not edge-reconstructible then \(G\) is \(k\)-free, for all even \(k\). To get closer, therefore, to settling the Edge-Reconstruction Conjecture, an investigation is begun into which graphs are, or are not, \(k\)-free (for different values of \(k\), in particular for \(k = 2\)); we also discuss which graphs are \(k\)-free, for all \(k\).

A \((v, k, \lambda)\) covering design of order \(v\), block size \(k\), and index \(\lambda\) is a collection of \(k\)-element subsets, called blocks of a set \(V\) such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks in a covering design. In this paper we solve the covering problem with \(k = 5\) and \(\lambda = 4\) and all positive integers \(v\) with the possible exception of \(v = 17, 18, 19, 22, 24, 27, 28, 78, 98\).

Let \(\rho\) be an \(h\)-ary areflexive relation. We study the complexity of finding a strong \(h\)-coloring for \(\rho\), which is defined in the same way for \(h\)-uniform hypergraphs.In particular we reduce the \(H\)-coloring problem for graphs to this problem, where \(H\) is a graph depending on \(\rho\). We also discuss the links of this problem with the problem of
finding a completeness criterion for finite algebras.

Let \( {Z}_k\) be the cyclic group of order \( k\). Let \( H\) be a graph. A function \( c: E(H) \to {Z}_k\) is called a \( {Z}_k\)-coloring of the edge set \( E(H)\) of \(H\). A subgraph \( G \subseteq H\) is called zero-sum (with respect to a \( {Z}_k\)-coloring) if \( \sum_{e \in E(G)} c(e) \equiv 0 \pmod{k}\). Define the zero-sum Turán numbers as follows. \( T(n, G, {Z}_k)\) is the maximum number of edges in a \( {Z}_k\)-colored graph on \( n\) vertices, not containing a zero-sum copy of \( G\). Extending a result of [BCR], we prove:

THEOREM.
Let \( m \geq k \geq 2\) be integers, \( k | m\). Suppose \( n > 2(m-1)(k-1)\), then
\[T(n,K_{1,m},{Z}_k) =
\left\{
\begin{array}{ll}
\frac{(m+k-2)-n}{2}-1, & if \;\; n-1 \equiv m \equiv k \equiv 0 \pmod{2}; \\
\lfloor \frac{(m+k-2)-n}{2} \rfloor, & otherwise.
\end{array}
\right.\]