The present paper studies bisectable trees, i.e., trees whose edges can be colored by two colors so that the induced monochromatic subgraphs are isomorphic. It is proved that the number of edges that have to be removed from a tree with maximum degree three to make it bisectable can be bounded by an absolute constant.

We study the maximal intersection number of known Steiner systems and designs obtained from codes. By using a theorem of Driessen, together with some new observations, we obtain many new designs.

Taking as blocks some subspace pairs in a finite unitary geometry, we construct a number of new Balanced Incomplete Block (BIB) designs and Partially Balanced Incomplete Block (PBIB) designs, and also give their parameters.

The support size of a factorization is the sum over the factors of the number of distinct edges in each factor. The spectrum of support sizes of \(l\lambda\)-factorizations of \(\lambda K_n\) and \(\lambda K_{n,n}\) are completely determined for all \(\lambda\) and \(n\).

Latin interchanges have been used to establish the existence of critical sets in Latin squares, to search for subsquare-free Latin squares, and to investigate the intersection sizes of Latin squares. Donald Keedwell was the first to study Latin interchanges in their own right. This paper builds on Keedwell’s work and proves new results about the existence of Latin interchanges.

A balanced ternary design of order nine with block size three, index two, and \(\rho_2 = 1\) is a collection of multi-subsets of size \(3\) (of type \(\{x, y, z\}\) or \(\{x, x, y\}\)) called blocks, chosen from a \(9\)-set, in which each unordered pair of distinct elements occurs twice, possibly in one block, and in which each element is repeated in just one block. So there are precisely \(9\) blocks of type \(\{x, x, y\}\). We denote such a design by \((9; 1; 3, 2)\) BTD. In this note, we describe the procedures we have used to
determine that there are exactly \(1475\) non-isomorphic \((9; 1; 3, 2)\) BTDs.