Let \(L\) be a linear form on the Galois field \(\text{GF}(q^{n+1})\) over \(\text{GF}(q)\) (\(n \geq 2\)). We characterize those integers \(s\) coprime to \(v = (q^{n+1}-1)/(q-1)\) such that \(L(x^s)\) is (or is related to) a quadratic form on \(\text{GF}(q^{n+1})\) over \(\text{GF}(q)\). This relates to a conjecture of Games concerning quadrics of the form \(rD\) in \(\text{PG}(n,q)\), where \(D\) is a difference set in the cyclic group \({Z}_v\), acting as a Singer group on the points and hyperplanes of \(\text{PG}(n,q)\). It has been shown that Games’ conjecture does not hold except possibly in the case \(q = 2\): here we establish that it holds exactly when \(q = 2\). We also suggest a new conjecture. Our result for \(q=2\) enables us to prove another conjecture of Games’, concerning \(m\)-sequences with three-valued periodic cross-correlation function.

Let \(G\) be a group acting on a set \(\Omega\). A subset (finite or infinite) \(A \subseteq \Omega\) is called \(k\)-quasi-invariant, where \(k\) is a non-negative integer, if \(|A^g \backslash A| \leq k\) for every \(g \in G\). In previous work of the authors a bound was obtained, in terms of \(k\), on the size of the symmetric difference between a \(k\)-quasi-invariant subset and the \(G\)-invariant subset of \(\Omega\) closest to it. However, apart from the cases \(k = 0, 1\), this bound gave little information about the structure of a \(k\)-quasi-invariant subset. In this paper a classification of \(2\)-quasi-invariant subsets is given. Besides the generic examples (subsets of \(\Omega\) which have a symmetric difference of size at most \(2\) with some \(G\)-invariant subset) there are basically five explicitly determined possibilities.

This note is an extension of \([4]\) , wherein is shown a relation between the dual notions of graceful and edge-graceful graphs. In particular, this note proves two graceful conjectures raised in \([4]\) , and then utilizes the result to edge-gracefully label certain trees not previously known to be edge-graceful.

We present sufficient conditions for the existence of a \(k\)-factor in a simple graph depending on \(\sigma_2(G)\) and the neighbourhood of independent sets in our first theorem and on \(\sigma_2(G)\) and \(\alpha(G)\) in the second one.

It is well known that a necessary condition for the existence of a \((v, 4, 1)\)-RPMD is \(v \equiv 0 \text{ or } 1 \pmod{4}\) and the existence of \((v, 4, 1)\)-RPMDs for \(v \equiv 1 \pmod{4}\) has been completely settled.
In this paper, we shall introduce the concept of \((v, k, 1)\)-nearly-RPMDs and use it to obtain some new construction methods for \((v, k, 1)\)-RPMDs with \(v \equiv 0 \pmod{k}\). As an application, we shall show that a \((v, 4, 1)\)-RPMD exists for all integers \(v \geq 4\) where \(v \equiv 0 \pmod{4}\), except for \(v = 4, 8\) and with at most \(49\) possible exceptions of which the largest is \(336\).
It is also well known that a \((v, k, 1)\)-RPMD exists for all sufficiently large \(v\) with \(k \geq 3\) and \(v \equiv 1 \pmod{k}\), and a \((v, k, 1)\)-PMD exists with \(v(v – 1) \equiv 0 \pmod{k}\) for the case when \(k\) is an odd prime and \(v\) is sufficiently large. In this paper, we shall show that there exists a \((v, k, 1)\)-RPMD for all sufficiently large \(v\) with \(v \equiv 0 \pmod{k}\), and there exists a \((v, k,\lambda)\)-PMD for all sufficiently large \(v\) with \(\lambda v(v – 1) \equiv 0 \pmod{k}\).