In this paper, we show that for every modular lattice \(L\), if its size is at least three times its excess, then each component of its direct product decomposition is isomorphic to one of the following: a Boolean lattice of rank one \(B_1\), a chain of length two \(3\), a diamond \(M_3\), and \(M_4\), where \(M_n\) is a modular lattice of rank two which has exactly \(n\) atoms.

Using algebraic curves, it will be proven that large partial unitals can be embedded into unitals and large \((k,n)\)-arcs into maximal arcs.

In a set equipped with a binary operation, \((S, \cdot)\), a subset \(U\) is defined to be avoidable if there exists a partition \(\{A, B\}\) of \(S\) such that no element of \(U\) is the product of two distinct elements of \(A\) or of two distinct elements of \(B\). For more than two decades, avoidable sets in the natural numbers (under addition) have been studied by renowned mathematicians such as Erdős, and a few families of sets have been shown to be avoidable in that setting. In this paper, we investigate the generalized notion of an avoidable set and determine the avoidable sets in several families of groups; previous work in this field considered only the case \((S, \cdot) = (\mathbb{N}, +)\).

This paper studied the problems of counting independent sets, maximal independent sets, and maximum independent sets of a graph from an algorithmic point of view. In particular, we present linear-time algorithms for these problems in trees and unicyclic graphs.

The Stirling numbers of first kind and Stirling numbers of second kind, denoted by \(s(n,k)\) and \(S(n,k)\) respectively, arise in a variety of combinatorial contexts. There are several algebraic and combinatorial relationships between them. Here, we state and prove four new identities concerning the determinants of matrices whose entries are unsigned Stirling numbers of first kind and Stirling numbers of second kind. We also observe an interrelationship between them based on our identities.

We generalize a construction by Treash of a Steiner triple system on \(2v+1\) points that embeds a Steiner triple system on \(v\) points. We show that any Steiner quadruple system on \(v+1\) points may be embedded in a Steiner quadruple system on \(2v+2\) points.

A \((\lambda K_n, G)\)-design is a partition of the edges of \(\lambda K_n\), into sub-graphs each of which is isomorphic to \(G\). In this paper, we investigate the existence of \((K_n, G_{16})\)-design and \((K_n, G_{20})\)-design, and prove that the necessary conditions for the existence of the two classes of graph designs are also sufficient.

Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \(ae\) is the absolute value of the difference of the labels of \(a\) and \(e\). A labeling of the vertices of a graph of order \(p\) is minimally \(k\)-equitable if the vertices are labeled with elements of \({1,2, \ldots, p}\) and in the induced labeling of its edges, every label either occurs exactly \(k\) times or does not occur at all. We prove that the corona graph \(C_{2n}OK_1\) is minimally \(4\)-equitable.

A set of Bishops cover a board if they attack all unoccupied squares. What is the minimum number of Bishops needed to cover an \(k \times n\) board \(?\) Yaglom and Yaglom showed that if \(k = n\), the answer is \(n\). We extend this result by showing that the minimum is \(2\lfloor \frac{n}{2}\rfloor\) if \(k 2k > 2\), a cover is given with \(2\lfloor\frac{k+n}{2}\rfloor\) Bishops. We conjecture that this is the minimum value. This conjecture is verified when \(k \leq 3\) or \(n \leq 2k + 5\).

It is proved that the following graphs are harmonious:(1) shell graphs (2) cycles with the maximum possible number of concurrent alternate chords (3) Some families of multiple shells