In this paper, the unimodality of \((r,\beta)\)-Stirling numbers and certain asymptotic approximation of \((r,\beta)\)-Bell numbers are established. Together with these results and the most general form of Central Limit Theorem, viz. Bounded Variance Normal Convergence Criterion, the \((r,\beta)\)-Stirling numbers are shown to be asymptotically normal.

The graph \(\mathcal{R}(d)\) of realizations of \(d\) is a graph whose vertices are the graphs with degree sequence \(d\), two vertices are adjacent in the graph \(\mathcal{R}(d)\) if one can be obtained from the other by a switching. It has been shown that the graph \(\mathcal{R}(d)\) is connected. Let \(\mathcal{CR}(d)\) be the set of connected graphs with degree sequence \(d\). Taylor \([13]\) proved that the subgraph of \(\mathcal{R}(d)\) induced by \(\mathcal{CR}(d)\) is connected. Several connected subgraphs of \(\mathcal{CR}(d)(3^n)\) are obtained in this paper. As an application, we are able to obtain the interpolation and extremal results for the number of maximum induced forests in the classes of connected subgraphs of \(\mathcal{CR}(d)(3^n)\).

Let \(T = (V, A)\) be a finite tournament with \(n \geq 2\) vertices. The dual of T is the tournament \(T^* = (V, A^*)\) defined by: for all \(x,y \in V, (x,y) \in A^*\) if and only if \((y,x) \in A\). The tournament \(T\) is critical if \(T\) is indecomposable and if for all \(x \in V\), the subtournament \(T(V – \{x\})\) is decomposable. A \(3\)-cycle is a tournament isomorphic to the tournament \(T, = ({0,1,2}, {(0, 1), (1, 2), (2, 0)})\). Let \(F\) be a set of non negative integers \(k < n\). The tournament \(T\) is \(F\)-selfdual if for every subset \(X\) of \(V\) such that \(|X |\in F\), the subtournaments \(T(X)\) and \(T^*(X)\) are isomorphic. In this paper, we study, for each integer \(k \geq 1\), the \(\{n – k\}\)-selfduality of the tournaments, with \(n \geq 4+k\) vertices, that are lexicographical sums of tournaments under a \(3\)-cycle or a critical tournament. As application, we determine for each integer \(k \geq 1\), the tournaments, with \(n \geq 4+ k\) vertices, that are \(\{4,n – k\}\)-selfdual.

This paper studies in detail the collection of closed sets of a matroid of arbitrary cardinality ordered by inclusion. The relation between the collection, in particular the collection of a simple matroid, and a finite length geometric lattice is dealt with. Finally, one obtains that up to isomorphism, a finite length geometric lattice is a simple matroid, and vice versa.

A vertex set \(D\) of a graph \(G\) is a dominating set if every vertex not in \(D\) is adjacent to some vertex in \(D\). The domination number \(\gamma\) of a graph \(G\) is the minimum cardinality of a dominating set in \(G\).

In 1975, Payan \([6]\) communicated without proof the inequality
\[2\gamma \leq {n} + 1 – \delta\]
for every connected graph not isomorphic to the complement of a one-regular graph, where \(n\) is the order and \(\delta\) the minimum degree of the graph. A first proof of (*) was published by Flach and Volkman \([3]\) in \(1980\).

In this paper, we firstly present a more transparent proof of (*). Using the idea of this proof, we show that

\[2\gamma \leq n – \delta\]

for connected graphs with exception of well-determined families of graphs.

We classify all finite linear spaces on at most \(15\) points admitting a blocking set. There are no such spaces on \(11\) or fewer points, one on \(12\) points, one on \(13\) points, two on \(14\) points, and five on \(15\) points. The proof makes extensive use of the notion of the weight of a point in a \(2\)-coloured finite linear space, as well as the distinction between minimal and non-minimal \(2\)-coloured finite linear spaces. We then use this classification to draw some conclusions on two open problems on the \(2\)-colouring of configurations of points.

Suppose \(G\) is a finite plane graph with vertex set \(V(G)\), edge set \(E(G)\), and face set \(F(G)\). The paper deals with the problem of labeling the vertices, edges, and faces of a plane graph \(G\) in such a way that the label of a face and labels of vertices and edges surrounding that face add up to a weight of that face. A labeling of a plane graph \(G\) is called \(d\)-antimagic if for every number \(s\), the \(s\)-sided face weights form an arithmetic progression of difference \(d\). In this paper, we investigate the existence of \(d\)-antimagic labelings for a special class of plane graphs.

The choice number of a graph \(G\), denoted by \(\chi_l(G)\), is the minimum number \(\chi_l\) such that if we give lists of \(\chi_l\) colors to each vertex of \(G\), there is a vertex coloring of \(G\) where each vertex receives a color from its own list no matter what the lists are. In this paper, we show that \(\chi_l(G) \leq 3\) for each plane graph of girth at least \(4\) which contains no \(8\)-circuits and \(9\)-circuits.

It is noted that Teirlinck’s “transposition argument” for disjoint \(\text{STS}(v)\) applies more generally to certain partial triple systems of different orders. A corollary on the number of blocks common to two \(\text{STS}(v)\) of different orders is also given.

We introduce a generalisation of the traditional magic square, which proves useful in the construction of magic labelings of graphs. An order \(n\) sparse semi-magic square is an \(n \times n\) array containing the entries \(1, 2, \ldots, m\) (for some \(m < n^2\)) once each with the remainder of its entries \(0\), and its rows and columns have a constant sum \(k\). We discover some of the basic properties of such arrays and provide constructions for squares of all orders \(n \geq 3\). We also show how these arrays can be used to produce vertex-magic labelings for certain families of graphs.