Let \( T \) be a partial Latin square. If there exist two distinct Latin squares \( M \) and \( N \) of the same order such that \( M \cap N = T \), then \( M \setminus T \) is said to be a Latin trade. For a given Latin square \( M \), it is possible to identify a subset of entries, termed a critical set, which intersects all Latin trades in \( M \) and is minimal with respect to this property.

Stinson and van Rees have shown that under certain circumstances, critical sets in Latin squares \( M \) and \( N \) can be used to identify critical sets in the direct product \( M \times N \). This paper presents a refinement of Stinson and van Rees’ results and applies this theory to prove the existence of two new families of critical sets.

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.