The trace of a degree \( n \) polynomial \( p(x) \) over \( \text{GF}(2) \) is the coefficient of \( x^{n-1} \), and the \({subtrace}\) is the coefficient of \( x^{n-2} \). We derive an explicit formula for the number of irreducible degree \( n \) polynomials over \( \text{GF}(2) \) that have a given trace and subtrace. The trace and subtrace of an element \( \beta \in \text{GF}(2^n) \) are defined to be the coefficients of \( x^{n-1} \) and \( x^{n-2} \), respectively, in the polynomial \(q(x) = \prod_{i=0}^{n-1} (x + \beta^{2^i}).\) We also derive an explicit formula for the number of elements of \( \text{GF}(2^n) \) of given trace and subtrace. Moreover, a new two-equation Möbius-type inversion formula is proved.

In this paper, it has been verified, by a computer-based proof, that the smallest size of a complete arc is 12 in \( \text{PG}(2,27) \) and 13 in \( \text{PG}(2,29) \). Also, the spectrum of the sizes of the complete arcs of \( \text{PG}(2,27) \) has been found. The classification of the smallest complete arcs of \( \text{PG}(2,27) \) is given: there are seven non-equivalent 12-arcs, and for each of them, the automorphism group and some geometrical properties are presented. Some examples of complete 13-arcs of \( \text{PG}(2,29) \) are also described.

For a factorization \( F \) of a graph \( G \) into factors \( F_1, F_2, \ldots, F_k \), the chromatic number \( \chi(F) \) of \( F \) is the minimum number of elements \( V_1, V_2, \ldots, V_m \) in a partition of \( V(G) \) such that each subset \( V_i \) \((1 \leq i \leq m)\) is independent in some factor \( F_j \) \((1 \leq j \leq k)\). If \( \chi(F) = m \), then \( F \) is an \( m \)-chromatic factorization. For integers \( k, m, n \geq 2 \) with \( n \geq m \), the cofactor number \( c_m(k,n) \) is defined as the smallest positive integer \( p \) for which there exists an \( m \)-chromatic factorization \( F \) of the complete graph \( K_p \) into \( k \) factors \( F_1, F_2, \ldots, F_k \) such that \( \chi(F_i) \geq n \) for all integers \( i \) \((1 \leq i \leq k)\). The values of the numbers \( c_m(k,n) \) are investigated for \( m = 3 \) and \( m = 4 \).The \( k \)-cofactorization number \( \chi_k(G) \) of a graph \( G \) is defined as \( \max\{\chi(F) : F \text{ is a factorization of } G \text{ into } k \text{ factors}\} \). It is shown that \( \chi_k(K_n) \geq \lfloor n^{1/k} \rfloor \) for \( k \geq 2 \) and \( n \geq 4 \). The numbers \( \chi_k(K_n) \) are determined for several values of \( k \) and \( n \).

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.

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.