We describe an algorithm for finding smallest defining sets of designs. Using this algorithm, we show that the 104 \(STS(19)\) which have automorphism group order at least 9 have smallest defining set sizes in the range 18-23. The numbers of designs with smallest defining sets of \(18, 19, 20, 21, 22\) and \(23\) blocks are, respectively, \(1, 2, 17, 68, 14\) and \(2\).

In this paper, three simple algorithms for the satisfiability problem are presented with their probabilistic analyses. One algorithm, called counting, is designed to enumerate all the solutions of an instance of satisfiability. The second one, namely E-SAT, is proposed for solving the corresponding decision problem. Both the enumeration and decision algorithms have a linear space complexity and a polynomial average time performance for a specified class of instances. The third algorithm is a randomized variant of E-SAT. Its probabilistic analysis yields a polynomial average time performance.

For any abelian group \*A\), we call a graph \(G = (V, E)\) as A-magic if there exists a labeling I: E(G) \(\to \text{A} – \{0\}\) such that the induced vertex set labeling \(I^+: V(G) \to A\)
\[\text{I}^+\text{(v)} = \Sigma \{ \text{I(u,v) : (u,v) in E(G)} \}\]
is a constant map. We denote the set of all \(A\) such that G is \(A\)-magic by \(AM(G)\) and call it as group-magic index set of \(G\).

Let \((\mathcal{P}, \mathcal{B}, \mathcal{I})\) be an asymmetric \((v, k, \lambda)\) block design. The incidence graph \(G\) of this design is distance-regular, hence belongs to an association scheme. In this paper, we use the algebraic structure of this association scheme to analyse certain symmetric partitions of the incidence structure.

A set with two intersection numbers is a subset \(\mathcal{K} \subseteq \mathcal{P}\) with the property that \(|{B} \cap \mathcal{K}|\) takes on only two values as \({B}\) ranges over the blocks of the design. In the special case where the design is a projective plane, these objects have received considerable attention. Two intersection theorems are proven regarding sets of this type which have a certain type of dual. Applications to the study of substructures in finite projective spaces of dimensions two and three are discussed.

In this paper, necessary and sufficient conditions for the existence of a 5-cycle system of the \(\lambda\)-fold complete graph of order \(v\) with a hole of size \(u\),\(\lambda(K_v – K_u)\), are proved.

Let \(G\) be a simple connected graph on \(2n\) vertices with a perfect matching. For a positive integer \(k\), \(1 \leq k \leq n – 1\), \(G\) is \(k\)-\({extendable}\) if for every matching \(M\) of size \(k\) in \(G\), there is a perfect matching in \(G\) containing all the edges of \(M\). For an integer \(k\), \(0 \leq k \leq n – 2\), \(G\) is \({strongly \;k-extendable}\) if \(G\) – \(\{u, v\}\) is \(k\)-extendable for every pair of vertices \(u\) and \(v\) of \(G\). The problem that arises is that of characterizing \(k\)-extendable graphs and strongly \(k\)-extendable graphs. The first of these problems has been considered by several authors whilst the latter has only been recently studied by the author. In a recent paper, we established a number of properties of strongly \(k\)-extendable graphs including some sufficient conditions for strongly \(k\)-extendable graphs. In this paper, we focus on a necessary condition, in terms of minimum degree, for strongly \(k\)-extendable graphs. Further, we determine the set of realizable values for minimum degree of strongly \(k\)-extendable graphs. A complete characterization of strongly \(k\)-extendable graphs on \(2n\) vertices for \(k = n – 2\) and \(n – 3\) is also established.

In this paper we discuss some designs that have been used to train mediators for dispute resolution and tabulate some small examples.

The spectrum \(Q(k,\lambda)\) of coset difference arrays has played an important role in Lu’s work on asymptotic existence of resolvable balanced incomplete block designs. In this article, we use Weil’s theorem on character sums to show that if \(k = 2\lambda + 1\), then for any prime power \(q \equiv 1+2k \pmod{4k}\), \(q \in Q(k,\lambda)\) whenever \(g > D(k) = (\frac{B+\sqrt{B^2+4C}}{2})^2\), where \(B = (k-2)k(2k-1)(2k)^{k-1} – (2k)^{k} + 1\) and \(C = \frac{(k-2)(k-1)}{2}(2k)^{k-1}\). In particular, we determine the spectrum \(Q(3,1)\). In addition, the degenerate case when \(k = \lambda + 1\) is also discussed.

The third author proved earlier [8] that if a Euclidean space is colored with red and blue so that the distance one is forbidden for blue, and translates of some \(k\)-point configuration are forbidden for red, then the unit-distance chromatic number of the space is no greater than \(k\). Here we give a generalization.

We continue the study of graphs defined by a certain adjacency property by investigating the \(n\)-existentially closed line-critical graphs. We classify the \(1\)-e.c. line-critical graphs and give examples of \(2\)-e.c. line-critical graphs for all orders \(\geq 9\).