We construct new simple \(3-(17,5,3), 3-(19,9,56), 3-(19,9,140)\), and \(3-(19,9,224)\) designs by combining disjoint designs.

An \(\text{NB}[k, \lambda; v]\) is a \(\text{B}[b, \lambda; v]\) which has no repeated blocks. In this paper we prove that there exists an indecomposable \(\text{NB}[3,5; v]\) for \(v \geq 7\) and \(v \equiv 1 \text{ or } 3 \pmod{6}\), with the exception of \(v = 7\) and \(9\), and the possible exception of \(v = 13, 15\).

We propose several invariants for cycle systems and \(2\)-factorizations of complete graphs, and enumerate the \(4\)- and \(6\)-cycle systems of \(K_g\).

Let \(G\) be a simple connected graph on \(2n\) vertices with a perfect matching. \(G\) is \(k\)-\({extendable}\) if for any set \(M\) of \(k\) independent edges, there exists a perfect matching in \(G\) containing all the edges of \(M\). \(G\) is \({minimally \; k-extendable}\) if \(G\) is \(k\)-extendable but \(G – uv\) is not \(k\)-extendable for every pair of adjacent vertices \(u\) and \(v\) of \(G\). The problem that arises is that of characterizing \(k\)-extendable and minimally \(k\)-extendable graphs. The first of these problems has been considered by several authors whilst the latter has only been recently studied. In a recent paper, we established several properties of minimally \(k\)-extendable graphs as well as a complete characterization of minimally \((n – 1)\)-extendable graphs on \(2n\) vertices. In this paper, we focus on characterizing minimally \((n – 2)\)-extendable graphs. A complete characterization of \((n – 2)\)-extendable and minimally \((n – 2)\)-extendable graphs on \(2n\) vertices is established.

We give the numbers of nonisomorphic \(2-(7,3,\lambda)\) block designs for \(\lambda = 6,7,8,9\). We discuss the method of generation and present statistics concerning automorphism groups and multiple blocks. The \(418\) \(2-(7, 3, 6)\) block designs together with the order of their automorphism groups are listed.

A union-closed family \(\mathcal{A} = \{A_1, A_2, \ldots, A_n\}\) is a non-empty finite collection of distinct non-empty finite sets, closed under union. It has been conjectured that for any such family, there is some element in at least half of its sets. But the problem remains unsolved. This paper establishes several results pertaining to this conjecture, with the main emphasis on the study of a possible counterexample with minimal \(n\).

The concept of the star chromatic number of a graph was introduced by Vince \([7]\), which is a natural generalization of the chromatic number of a graph. In this paper, we will prove that if the complement of a graph \(G\) is disconnected, then its star chromatic number is equal to its chromatic number. From this, we derive a number of interesting results. Let \(G\) be a graph such that the product of its star chromatic number and its independence ratio is equal to \(1\). Then for any graph \(H\), the star chromatic number of the lexicographic product of graphs \(G\) and \(H\) is equal to the product of the star chromatic number of \(G\) and the chromatic number of \(H\). In addition, we present many classes of graphs whose star chromatic numbers are equal to their chromatic numbers.

We obtain a formula for the number of finite lattices of a given height and cardinality that have a series-parallel and interval order. Our approach is to consider a naturally defined class of \(m\) nested intervals on an \(m+k\)-element chain, and we show that there are \(\binom{m-1}{k-1}\gamma(m+1)\) such sets of nested intervals. Here, \(\gamma(m+1)\) denotes the Catalan number \(\frac{1}{m+1}\binom{2m}{m}\).

\({Graph \; integrity}\), a measure of graph vulnerability, has drawn considerable attention of graph theorists in recent years. We have given a set of sufficient conditions for the weighted integrity problem to be NP-Complete for a class of graphs. As a corollary of this result, we have shown that the weighted graph integrity problem is NP-Complete on many common classes of graphs on which the unweighted integrity problem is either polynomial or of unknown complexity. We have shown that the weighted graph integrity problem is polynomial for interval graphs.