If \(\alpha\) is a primitive root of the finite field \({GF}(2^n)\), we define a function \(\pi_n\) on the set \({E}_n = \{1, 2, \ldots, 2^n – 2\}\) by
\[
\pi_\alpha(i) = j \quad \text{iff} \quad \alpha^i = 1 + \alpha^{j}.
\]
Then \(\pi_\alpha\) is a permutation of \({E}_n\) of order \(2\). The path-length of \(\pi\), denoted \({PL}(\pi)\), is the sum of all the quantities \(|\pi(i) – i|\),
and the rank of \(\pi\) is the number of pairs \((i, j)\) with \(i \pi(j)\). We show that \({PL}(\pi) = {2(2^n – 1)(2^{n-1} – 1)}/{3}\), and the rank of \(\pi\) is \((2^{n-1} – 1)^2\).

If \(\gcd(k, 2^n – 1) = 1\), then \(M_k(x) = kx(\mod{2^n – 1})\) is a permutation of \({E}_n\). We show that a necessary condition for the function \(f_i(x) = 1 + x + \cdots + x^{i}\) to be a permutation of \({GF}(2^n)\), is that the function \(g_k(r) = \pi(M_{k+1}(r)) – \pi(r)\) be a permutation of \({E}_n\) such that exactly half the members \(r\) of \({E}_n\) satisfy \(g_k(r) r\).

Let \((G, \cdot)\) be a group with identity element \(e\) and with a unique element \(h\) of order \(2\). In connection with an investigation into the admissibility of linear groups, one of the present authors was recently asked if, for every cyclic group \(G\) of even order greater than \(6\), there exists a bijection \(\gamma\)
from \(G \setminus \{e, h\}\) to itself such that the mapping \(\delta: g \to g \cdot \gamma(g)\) is again a bijection from \(G \setminus \{e, h\}\) to itself. In the present paper, we answer the above question in the affirmative and we prove the
more general result that every abelian group which has a cyclic Sylow \(2\)-subgroup of order greater than \(6\) has such a partial bijection.

A directed triple system of order \(v\) and index \(\lambda\),denoted \({DTS}_\lambda(v)\), is said to be reverse if it admits an automorphism consisting of \(v/2\) transpositions when \(v\) is even, or a fixed point and \((v-1)/2\) transpositions when \(v\) is odd. We give necessary and sufficient conditions for the existence of a reverse \({DTS}_\lambda(v)\) for all \(\lambda \geq 1\).

A \(1\)-\emph{factor} of a graph \(G\) is a \(1\)-regular spanning subgraph of \(G\).A graph \(G\) has exactly \(t\) \(1\)-factors if the maximum set of edge-disjoint \(1\)-factors is \(t\). For given non-negative integers \(d\), \(t\), and even \(e\), let \(\mathcal{G}(2n; d, e, t)\) be the class of simple connected graphs on \(2n\) vertices, \((2n-1)\) of which have degree \(d\) and one has degree \(d+e\),having exactly \(t\) \(1\)-factors. The problem that arises is that of determining when \(\mathcal{G}(2n; d, e, t) \neq \emptyset ?\) Recently, we resolved the case \(t = 0\). In this paper, we will consider the case \(t = 1\).

In this paper we show that the complete graph \(K_{12}\) is not decomposable into three factors of diameter two, thus resolving a longstanding open problem. This result completes the solution of decomposition of a complete graph into three
factors, one of which has diameter two and the other factors have finite diameters.

The edge-integrity of a graph measures the difficulty of breaking it into pieces through the removal of a set of edges, taking into account both the number of edges removed and the size of the largest surviving component. We develop some techniques for bounding, estimating and computing the edge-integrity of products of graphs, paying particular attention to grid graphs.

We describe an algorithm for computing the number \(h_{m,n}\) of Hamiltonian circuits in a rectangular grid graph consisting of \(m \times n\) squares. For any fixed \(m\), the set of Hamiltonian circuits on such graphs (for varying \(n\)) can be identified via an appropriate coding with the words of a certain regular language \(L_m \subset (\{0,1\}^m)^*\). We show how to systematically construct a finite automaton \(A_m\) recognizing \(L_m\). This allows, in principle, the computation of the (rational) generating function \(h_m(z)\) of \(L_m\). We exhibit a bijection between the states of \(A_m\) and the words of length \(m+1\) of the familiar Motzkin language. This yields an upper bound on the degree of the denominator polynomial of \(h_m(z)\), hence of the order of the linear recurrence satisfied by the coefficients \(h_{m,n}\) for fixed \(m\).

The method described here has been implemented. Numerical data resulting from this implementation are presented at the end of this article.

The core \(G_\Delta\) is the subgraph of \(G\) induced by the vertices of maximum degree. If \(\Delta(G)\) is a forest, then Fournier [8] showed that \(G\) is Class 1. Therefore, if \(G\) is Class 2, \(G_\Delta\) must contain cycles. A natural question is this: what is the chromatic index of \(G\) if \(G_\Delta\) is a union of vertex-disjoint cycles? In this paper, we give further information about this question.

A \(k\)-URD\((v, g, r)\) is a resolvable design on \(v\) points with block sizes \(g\) and \(k\). Each parallel class contains only one block size, and there are \(r\) parallel classes with blocks of size \(g\), this implies there are \(\frac{v-1-r(g-1)}{k-1}\) parallel classes of size \(k\).We show that for sufficiently large \(v\), the necessary conditions are sufficient for the following range of values of \(r\). Let \(\epsilon_{k,g} = 1\)if \(g \equiv 0 \mod{k}\) and \(k\) otherwise, and let \(u = \frac{v}{g\epsilon_{k,g}}\).If \(k = 2\) for all \(g\), or \(k = 3\) with \(g\) odd, then there exists a \(k\)-URD\((v, g, r)\) for the following values of \(r\):

1. If \(u\) is an odd prime power, then for all \(1 \leq r \leq \frac{1}{k-1}(u-1)\), except possibly for the case where \(k = 3\) and \(u\) is congruent to \(5 \mod 6\).
2. If \(u\) is not prime, then for all \(1 \leq r \leq \frac{u}{p}\), where \(p\) is the smallest proper factor of \(u\).

Motivated by questions about semilattices of ordered compactifications, we study the structure of the lattice \(\mathcal{Q}(Y)\) of all closed quasiorders on a (compact) Hausdorff space \(Y\). For example, we show that the meets of coatoms are precisely those quasiorders which make the underlying space totally order-disconnected.
We describe the covering relation of such lattices and characterize “modular” and “semimodular” elements. In particular, we show that the closed equivalence relations on \(Y\) are precisely those upper semimodular elements of \(\mathcal{Q}(Y)\) which are not coatoms, and for \(|Y| \geq 3\), they are just the joins of bi-semimodular elements.
As a consequence of these results, two compact spaces are homeomorphic if and only if their lattices of closed quasiorders are isomorphic.