It is well known that there exist complete \(k\)-caps in \(\mathrm{PG}(3,q)\) with \(k \geq \frac{q^2+q+4}{2}\) and it is still unknown whether or not complete \(k\)-caps of size \(k < \frac{q^2+q+4}{2}\) and \(q\) odd exist. In this paper sufficient conditions for the existence of complete \(k\)-caps in \(\mathrm{PG}(3,q)\), for good \(q \geq 7\) and \(k < \frac{q^2+q+4}{2}\), are established and a class of such complete caps is constructed.

It is proved in this paper that for any given odd integer \(\lambda \geq 1\), there exists an integer \(v_0 = v_0(\lambda)\), such that for \(v > v_0\), the necessary and sufficient conditions for the existence of an indecomposable triple system \(B(3,\lambda; v)\) without repeated blocks are \(\lambda(v – 1) \equiv 0 \pmod{2}\) and \(\lambda{v(v – 1)} \equiv 0 \pmod{6}.\)

We prove that if \(G\) is a 1-tough graph with \(n = |V(G)| \geq 13\) such that
the degree sum of any three independent vertices is at least \(\frac{3n-14}{2}\), then \(G\) is hamiltonian.

This paper deals with the problem of labeling the vertices, edges, and interior faces of a grid graph in such a way that the label of the face itself and the labels of vertices and edges surrounding that face add up to a value prescribed for that face.

Let \(G\) be a 3-edge-connected simple triangle-free graph of order \(n\) . Using a contraction method, we prove that if \(\delta(G) \geq 4\) and if \(d(u) + d(v) > n/10\) whenever \(uv \in E(G)\) (or whenever \(uv \notin E(G)\) ), then the graph \(G\) has a spanning eulerian sub-
graph. This implies that the line graph \(L(G)\) is hamiltonian. We shall also characterize the extremal graphs.

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 \(k\).