We obtain necessary conditions for the enclosing of a group divisible design with block size 3, \( \text{GDD}(n, m; \lambda) \), into a group divisible design \( \text{GDD}(\text{n}, \text{m+1}; \lambda+\text{x}) \) with one extra group and minimal increase in \( \lambda \). We prove that the necessary conditions are sufficient for the existence of all such enclosings for GDDs with group size 2 and \( \lambda \leq 6 \), and for any \( \lambda \) when \( v \) is sufficiently large relative to \( \lambda \).

A known convolution identity involving the Catalan numbers is presented and discussed. Catalan’s original formulation, which is algebraically straightforward, is similar in style to one reported previously by the first author and the result has some interesting combinatorial aspects.

In this paper we prove various properties of the meanders. We then use these properties in order to construct recursively the set of all meanders of any particular order.

The main results of this paper are the discovery of infinite families of flow equivalent pairs of \( B_5 \) and \( W_5 \), amalamorphs, and infinite families of chromatically equivalent pairs of \( P \) and \( W_5^* \); homeomorphs, where \( B_5 \) is \( K_5 \) with one edge deleted, \( P \) is the Prism graph, and \( W_5 \) is the join of \( K_1 \) and a cycle on 4 vertices. Six families of \( B_5 \) amalamorphs, with two families having 6 parameters, and 9 families of \( W_5 \) amalamorphs, with one family having 4 parameters, are discovered. Since \( B_5 \) and \( W_5 \) are both planar, all these results obtained can be stated in terms of chromatically equivalent pairs of \( B_5^* \) and \( W_5^* \) homeomorphs. Also, three conjectures are made about the non-existence of flow-equivalent amalamorphs or chromatically equivalent homeomorphs of certain graphs.

Agrawal provided a construction for designs for two-way elimination of heterogeneity, based on a symmetric balanced incomplete block design. He could not prove the construction, although he found no counterexample. Subsequently Raghavarao and Nageswarerao published a proof of the method. In this note we observe a flaw in the published proof.

We discuss van der Waerden’s theorem on arithmetic progressions and an extension using Ramsey’s theorem, and the canonical versions. We then turn to a result (Theorem 6 below) similar in character to van der Waerden’s theorem, applications of Theorem 6, and possible canonical versions of Theorem 6. We mention several open questions involving arithmetic progressions and other types of progressions.

Let \( c^* = \). If we remove the double edge, the result is a \( 4 \)-cycle. Let \( (S,T) \) be a \( 2 \)-fold triple system without repeated triples and \( (S,C^*) \) a pairing of the triples into copies of \( c^* \). If \( C \) is the collection of \( 4 \)-cycles obtained by removing the double edges from each copy of \( c^* \) and \( F \) is a reassembly of these double edges into \( 4 \)-cycles, then \( (S,C \cup F) \) is a \( 2 \)-fold \( 4 \)-cycle system. We show that the spectrum for \( 2 \)-fold triple systems having a \({metamorphosis}\) into a \( 2 \)-fold \( 4 \)-cycle system as described above is precisely the set of all \( n \equiv 0,1,4\, \text{or}\, 9 \pmod{12} \geq 5 \).

Consider a graph \( G \) in which the vertices are partitioned into \( k \) subsets. For each subset, we want a set of vertices of \( G \) that dominate that subset. Note that the vertices doing the domination need not be in the subset itself. We are interested in dominating the entire graph \( G \) as well as dominating each of the \( k \) subsets and minimizing the sum of these \( k + 1 \) dominating sets. For trees and for all values of \( k \), we can determine an upper bound on this sum and characterize the trees that achieve it.

A technique is described that constructs a 4-colouring of a planar triangulation in quadratic time. The method is based on iterating Kempe’s technique. The heuristic gives rise to an interesting family of graphs which cause the algorithm to cycle. The structure of these graphs is described. A modified algorithm that appears always to work is presented. These techniques may lead to a proof of the 4-Colour Theorem which does not require a computer to construct and colour irreducible configurations.

For \( k > 0 \), we call a graph \( G = (V,E) \) \( k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k^* \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \), defined by \(l^+(v) = \sum\{l(u,v): (u,v) \in E(G)\}\) is a constant map. We denote the set of all \( k \) such that \( G \) is \( k \)-magic by \(\text{IM}(G)\). We call this set the \textbf{\emph{integer-magic spectrum}} of \( G \). We investigate these sets for trees, double trees, and abbreviated double trees. We define group-magic spectrum for \( G \) similarly. Finally, we show that a tree is \( k \)-magic, \( k > 2 \), if and only if it is \( k \)-label reducible.