We examine permutations having a unique fixed point and a unique reflected point; such permutations correspond to permutation matrices having exactly one \(1\) on each of the two main diagonals. The permutations are of two types according to whether or not the fixed point is the same as the reflected point. We also consider permutations having no fixed or reflected points; these have been enumerated using two different methods, and we employ one of these to count permutations with unique fixed and reflected points.

Let \(G\) be a graph and \(t(G)\) be the number of triangles in \(G\). Define \(\mathcal G_n\) to be the set of all graphs on \(n\) vertices that do not contain a wheel and \(t_n = \max\{t(G) : G \in \mathcal G_n\}\).
T. Gallai conjectured that \(t_n \leq \lfloor\frac{n^2}{8}\rfloor\). In this note we describe a graph on \(n\) vertices that contains no wheel and has at least \(\frac{n^2+n}{8}-3\) triangles.

A construction is given which uses \(\text{PG}_i(d, q)\) and \(q\) copies of \(\text{AG}_i(d, q)\) to construct designs having the parameters of \(\text{PG}_{i+1}(d+1, q)\), where \(q\) is a prime power and \(i \leq d-1\).

In a previous paper, [6], we associated with every hyperoval of a projective plane of even order a Hadamard \(2\)–design and investigated when this design has lines with three points. We study further this problem using the concept of regular triple and prove the existence of lines with three points in Hadamard designs associated with translation hyperovals. In the general case, the existence of a secant line of regular triples implies that the order of the projective plane is a power of two.

An incomplete self-orthogonal latin square of order \(v\) with an empty subarray of order \(n\), an ISOLS(\(v, n\)), can exist only if \(v \geq 3n+1\). It is well known that an ISOLS(\(v, n\)) exists whenever \(v \geq 3n+1\) and \((v,n) \neq (6m+2,2m)\). In this paper we show that an ISOLS(\(6m+2, 2m\)) exists for any \(m \geq 24\).

Graceful and edge-graceful graph labelings are dual notions of each other in the sense that a graceful labeling of the vertices of a graph \(G\) induces a labeling of its edges, whereas an edge-graceful labeling of the edges of \(G\) induces a labeling of its vertices. In this paper we show a connection between these two notions, namely, that the graceful labeling of paths enables particular trees to be labeled edge-gracefully. Of primary concern in this enterprise are two conjectures: that a path can be labeled gracefully starting at any vertex label, and that all trees of odd order are edge-graceful. We give partial results for the first conjecture and extend the domain of trees known to be edge-graceful for the second conjecture.

In this paper we establish a number of new lower bounds on the size of a critical set in a latin square. In order to do this we first give two results which give critical sets for isotopic latin squares and conjugate latin squares. We then use these results to increase the known lower bound for specific classes of critical sets. Finally, we take a detailed look at a number of latin squares of small order. In some cases, we achieve an exact lower bound for the size of the minimal critical set.

We characterize “effectively” all greedy ordered sets, relative to the jump number problem, which contain no four-cycles. As a consequence, we shall prove that \(O(P) = G(P)\) \quad whenever \(P \text{ greedy ordered set with no four-cycles.}\)

This paper sketches the method of studying the Multiplier Conjecture that we presented in [1], and adds one lemma. Applying this method, we obtain some partial solutions for it: in the case \(v = 2n_1\), the Second Multiplier Theorem holds without the assumption ”\(n_1 > \lambda\)”, except for one case that is yet undecided where \(n_1\) is odd and \(7 \mid \mid v\) and \(t \equiv 3, 5,\) or \(6 \pmod{7}\), and for every prime divisor \(p (\neq 7)\) of \(v\) such that the order \(w\) of \(2\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\); in the case \(n = 3n_1\) and \((v, 3 . 11) = 1\), then the Second Multiplier Theorem holds without the assumption “\(n_1 > \lambda\)” except for one case that is yet undecided where \(n_1\) cannot divide by \(3\) and \(13 \mid \mid v\) and the order of \(t\) mod \(13\) is \(12, 4,\) or \(6,2\), and for every prime divisor \(p (\neq 13)\) of \(v\) such that the order \(w\) of \(3\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\). These results distinctly improve McFarland’s corresponding results and Turyn’s result.