Halberstam, Hoffman and Richter introduced the idea of a Latin triangle as an analogue of a Latin square, showed the existence or non-existence of Latin triangles for small orders, and used a multiplication technique to generate triangles of orders \(3^n\) and \(3^n – 1\). We generalize this multiplication theorem and provide a construction of Latin triangles of odd order \(n\) for \(n\) such that \(n+2\) is prime. We also discuss scalar multiplication, orthogonal triangles, and results of computer searches.

A graph covering projection is a local graph homeomorphism. Certain partitions of the vertex set of the preimage graph induce a notion of “concreteness”. The concrete graph covering projections will be counted up to isomorphism.

The set of all distinct blocks of a \(t\)-design is referred to as the support of the design and its cardinality is denoted by \(b^*\). In this article (i) the set of all possible \(b^*\)’s for the case of \(3\)-\((8,4,\lambda)\) designs is determined and for each feasible \(b^*\) a design with a minimum \(b\) is produced;(ii) it is shown that a \(2\)-\((8,4,3\lambda)\) design is a \(3\)-\((8,4,\lambda)\) design if and only if it is self-complementary; (iii) it is shown that there are at least \(63\) pairwise non-isomorphic \(3\)-\((8,4,5)\) designs.

The cycle graph \(C(G)\) of a graph \(G\) has vertices which correspond to the chordless cycles of \(G\), and two vertices of \(C(G)\) are adjacent if the corresponding chordless cycles of \(G\) have at least one edge in common. If \(G\) has no cycle, then we define \(C(G)=\emptyset\), the empty graph. For an integer \(n \geq 2\), we define recursively the \(n\)-th iterated cycle graph \(C^n(G)\) by \(C^n(G)=C(C^{n-1}(G))\). We classify graphs according to their cycle graphs as follows. A graph \(G\) is \emph{cycle-vanishing} if there exists an integer \(n\) such that \(C^n(G)=\emptyset\); and \(G\) is \emph{cycle-periodic} if there exist two integers \(n\) and \(p \geq 1\) such that \(C^{n+p}(G)\cong C^n(G) \neq \emptyset\). Otherwise, \(G\) is cycle-expanding. We characterize these three types of graphs, and give some other results on cycle graphs.

A Latin square of order \(n\) is an \(n \times n\) array such that each of the integers \(1, 2, \ldots, n\) (or any set of \(n\) distinct symbols) occurs exactly once in each row and each column. A Latin square \(L = [l_{i,j}]\) is said to be \underline{commutative} provided that \(l_{i,j} = l_{j,i}\) for all \(i\) and \(j\). Two Latin squares, \(L = [l_{i,j}]\) and \(M = [m_{i,j}]\), are said to have \underline{intersection} \(k\) if there are exactly \(k\) cells \((i,j)\) such that \(l_{i,j} = m_{i,j}\).

Let \(I[n] = \{0, 1, 2, \ldots, n^2-9, n^2-8, n^2-7, n^2-6,n^2\}\), \(H[n] = I[n] \cup \{n^2-7, n^2-4\}\), and \(J[n]\) be the set of all integers \(k\) such that there exists a pair of commutative Latin squares of order \(n$ which have intersection \(k\). In this paper, we prove that \(J[n] = I[n]\) for each odd \(n \geq 7\), \(J[n] = H[n]\) for each even \(n \geq 6\), and give a list of \(J[n]\) for \(n \leq 5\). This totally solves the intersection problem of two commutative Latin squares.

Golomb and Taylor (joined later by Etzion) have modified the notion of a complete Latin square to that of a Tuscan-\(k\) square. A Tuscan-\(k\) square is a row Latin square with the further property that for any two symbols \(a\) and \(b\) of the square, and for each \(m\) from \(1\) to \(k\), there is at most one row in which \(b\) is the \(m^{th}\) symbol to the right of \(a\). One question unresolved by a series of papers of the authors mentioned was whether or not \(n \times n\) Tuscan-\(2\) squares exist for infinitely many composite values of \(n+1\). It is shown here that if \(p\) is a prime and \(p \equiv 7 \pmod{12}\) or \(p \equiv 5 \pmod{24}\), then Tuscan-\(2\) squares of side \(2p\) exist. If \(p \equiv 7 \pmod{12}\), clearly \(2p + 1\) is always composite and if \(p \equiv 5 \pmod{24}\), \(2p+1\) is composite infinitely often. The squares constructed are in fact Latin squares that have the Tuscan-\(2\) property in both dimensions.

It is shown that if \([n] = X_1 \cup X_2 \cup \cdots \cup X_l\) is a partition of \([n]\) and if \(S_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(k\) (circularly) consecutive blocks, then \(|S_t| < t^{n-k}\). If given \(a_1 < a_2 < \cdots < a_y \leq l \), \(\acute{S}_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(X_{a_{1}+m}, X_{a_{2}+m}, \ldots, X_{a_{k}+m}\) for some \(m\) with \(1-a_1 \leq m \leq n – a_k\), then \(|\acute{S}_t| \leq t^{n-k}\). Both these results were conjectured by Faudree, Schelp, and Sós [FSS]. The main idea of our proofs is that of anticlusters introduced by Griggs and Walker [GW] which we discuss in some detail. We also discuss several related intersection theorems about sets, \(2\)-valued functions, and \(t\)-valued functions.

Let the edges of the complete graph \(K_n\) be \(2\)-colored. A Simple Hamiltonian Cycle is a Hamiltonian cycle in \(K_n\) that is either monochromatic or is a union of two monochromatic paths. The main result of this paper is that if \(n\) is an even integer greater than \(4\), then for every \(2\)-coloring of the edges of \(K_n\), there is a Simple Hamiltonian Cycle in \(K_n\) which is either monochromatic, or is a union of two monochromatic paths, where each path is of even length.

Uniquely pseudointersectable graphs are defined; this is closely related to the uniquely intersectable graphs introduced by Alter and Wang [1]. The S-property is necessary but not sufficient for a graph to be uniquely pseudointersectable. This condition is also sufficient for graphs with unique minimum cover. Finally, we show that for supercompact graphs, unique pseudointersectability and unique intersectability are equivalent. Thus we generalize some of the results in [1] to a wider class of graphs.