The queen’s graph \(Q_n\) has the squares of the \(n \times n\) chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let \(\gamma(Q_n)\) be the minimum size of a dominating set of \(Q_n\). Spencer proved that \(\gamma(Q_n) \geq {(n-1)}/{2}\) for all \(n\), and the author showed \(\gamma(Q_n) = {(n-1)}/{2}\) implies \(n \equiv 3 \pmod{4}\) and any minimum dominating set of \(Q_n\) is independent.

Define a sequence by \(n_1 = 3\), \(n_2 = 11\), and for \(i > 2\), \(n_i = 4n_{i-1} – n_{i-2} – 2\). We show that if \(\gamma(Q_n) = {(n-1)}/{2}\) then \(n\) is a member of the sequence other than \(n_3 = 39\), and (counting from the center) the rows and columns occupied by any minimum dominating set of \(Q_n\) are exactly the even-numbered ones. This improvement in the lower bound enables us to find the exact value of \(\gamma(Q_n)\) for several \(n\); \(\gamma(Q_n) = {(n+1)}/{2}\) is shown here for \(n = 23, 39\), and elsewhere for \(n = 27, 71, 91, 115, 131\).

A characterization of symmetric bent functions has been presented in [3]. Here, we provide a simple proof of the same result.

We prove that the total domination number of an \(n\)-vertex claw-free cubic graph is at most \({n}/{2}\).

This paper deals with the problem of labeling the edges of a plane graph in such a way that the weight of a face is the sum of the labels of the edges surrounding that face. The paper describes \((a, d)\)-face antimagic labeling of a certain class of convex polytopes.

Below, we prove that there are exactly 244 nonisomorphic cyclic decompositions of the complete graph \(K_{25}\) into cubes. The full list of such decompositions is given in the Appendix.

The magic square is probably the most popular and well-studied topic in recreational mathematics. We investigate a variation on this classic puzzle — the antimagic square. We review the history of the problem, and the structure of the design. We then present computational results on the enumeration and construction. Finally, we describe a construction for all orders.

We establish a necessary and sufficient condition for the existence of a perfect distance-\(d\) placement in 3-dimensional tori, for both regular and irregular cases.