The number \(g^{(4)}_{2}\) is the minimal number of blocks that contain all pairs from a set of \(8\) elements exactly twice under the restriction that the longest block has size \(4\) (this longest block need not be unique). Thus the blocks have lengths \(2, 3\), and \(4\). We show that there are three solutions to this problem.

The \(n \times n\) primitive nearly reducible Boolean matrices whose \(k\)-exponents (\(1 \leq k \leq n\)) achieve the maximum value are characterized.

A graph is said to be \(k\)-covered if for each edge \(xy\), \(deg(x) = k\) or \(deg(y) = k\). In this paper, we characterize the \(3\)-covered quadrangulations of closed surfaces.

A graceful graph with \(n\) edges and \(n+1\) vertices is called a vertex-saturated graph. Each graceful graph corresponds to a vertex-saturated graph. Four classes of graceful graphs associated with vertex-saturated graphs are presented. Three of which generalize the results of [1], [2] and [5].

We correct an earlier theorem and reprove its consequences regarding \(c\)-BRDs with \(v \equiv 5, 8 \pmod{12}\). The original conclusions remain valid.

The type of a vertex \(v\) in a \(p\)-page book-embedding is the \(p \times 2\) matrix of nonnegative integers

\[{r}(v) =
\left(
\begin{array}{ccccc}
l_{v,1} & r_{v,1} \\
. & . \\
. & . \\
. & . \\
l_{v,p} & r_{v,p} \\
\end{array}
\right),\]

where \(l_{v,i}\) (respectively, \(r_{v,i}\)) is the number of edges incident to \(v\) that connect on page \(i\) to vertices lying to the left (respectively, to the right) of \(v\). The type number of a graph \(G\), \(T(G)\), is the minimum number of different types among all the book-embeddings of \(G\). In this paper, we disprove the conjecture by J. Buss et al. which says for \(n \geq 4\), \(T(L_n)\) is not less than \(5\) and prove that \(T(L_n) = 4\) for \(n \geq 3\).

Let \(T\) be a chemical tree, i.e. a tree with all vertices of degree less than or equal to \(4\). We find relations for the \(0\)-connectivity and \(1\)-connectivity indices \({}^0\chi(T)\) and \({}^1\chi(T)\), respectively, in terms of the vertices and edges of \(T\). A comparison of these relations with the coefficients of the characteristic polynomial of \(T\) associated to its adjacency matrix is established.

Given a regular action of a finite group \(G\) on a set \(V\), we consider the problem of the existence of an incidence structure \(\mathcal{I} = (V, \mathcal{B})\) on the set \(V\) whose full automorphism group \(Aut(\mathcal{I})\) is the group \(G\) in its regular action. Using results on graphical and digraphical regular representations \(([2,7], [1])\), we show the existence of such an incidence structure for all but four small finite groups.

For a finite field \({F} = {F}(q)\), where \(q = p^n\) is a prime power, we will introduce the notion of equivalence of subsets of \(F\) which stems out of the equivalence of cyclic difference sets, and give the formulae for the number of equivalence classes of \(k\)-subsets of \(F\) as well as for the number of equivalence classes of subsets of \(F\) by using Pólya’s theorem of counting.