It is proved that the \(n\)-cone \(C_m \vee K_n^c\) is graceful for any \(n \geq 1\) and \(m = 0\) or \(3 \pmod{12}\). The gracefulness of the following \(n\)-cones is also established: \(C_4 \vee K_n^c\), \(C_5 \vee K_2^c\), \(C_7 \vee K_n^c\), \(C_9 \vee K_2^c\), \(C_{11} \vee K_n^c\), \(C_{19} \vee K_n^c\). This partially answers the question of gracefulness of \(n\)-cones which is listed as an open problem in the survey article by J.A. Gallian.

We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to find tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the difficulty of obtaining a general result may be related to different behaviours of the capacity, depending on the “structure” of the odd integer representing the cycle length. We also describe the outcomes of some experiments, from which we derive the evidence that the Shannon capacity of odd cycles is extremely close to the value of the Lovasz theta function.

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.