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 \(\geq 9\).

An isometric path is merely any shortest path between two vertices. Inspired by the game of `Cops and Robber’ and a result by Aigner \(\&\) Fromme [1], we are interested in determining the minimum number of isometric paths required to cover the vertices of a graph. We find a lower bound on this number in terms of the diameter of a graph and find the exact number for trees and grid graphs.

An edge-graceful \((p, q)\)-graph \(G = (V, E)\) is a graph with \(p\) vertices and \(q\) edges for which there is a bijection \(f : E \to \{1,2,\ldots,q\}\) such that the induced mapping \(f^+ : V \to \mathbb{Z}_p\), defined by \(f^+(u) \equiv \sum\limits_{uv \in E} f(uv) \pmod{p}\), for \(u \in V\), is a bijection. In this paper, some results on edge-gracefulness of trees are extended to \(k\)-fold graphs based on graphs with \(p\) vertices and \(p – 1\) edges. A \(k\)-fold multigraph \(G[k]\) derived from a graph \(G\) is one in which each edge of \(G\) has been replaced by \(k\) parallel edges with the same vertices as the original edge. Certain classes of \(k\)-fold multigraphs derived from paths, combs, and spiders are shown to be edge-graceful, as well as other graphs constructed by combining these graphs in specified ways.

We determine solutions to the problem of gossiping in minimum time (briefly: minimum time problem or MTP) which require less calls than the previously known solutions for infinitely many values of the number \(n\) of persons and optimal solutions to the MTP, i.e. solutions of the MTP which minimize the number of calls, for some values of \(n\). We conjecture that our methods provide optimal solutions of the MTP for all \(n\).