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.

Let \(G\) be a simple graph and \(f\) a function from the vertices of \(G\) to the set of positive integers. An \((f, n)\)-coloring of \(G\) is an assignment of \(n\) colors to the vertices of \(G\) such that each vertex \(x\) is adjacent to less than \(f(x)\) vertices with the same color as \(x\). The minimum \(n\) such that an \((f, n)\)-coloring of \(G\) exists is defined to be the \(f\)-chromatic number of \(G\). In this paper, we address a study of this kind of locally restricted coloring.

A \(G\)-decomposition of the complete graph \(K_v\) is a set \({S}\) of subgraphs of \(K_v\), each isomorphic to \(G\), such that the edge set of \(K_v\) is partitioned by the edge sets of the subgraphs in \({S}\). For all positive integers \(v\) and every 2-regular graph \(G\) with ten or fewer vertices, we prove necessary and sufficient conditions for the existence of a \(G\)-decomposition of \(K_v\).

A broadcast graph on \(n\) vertices is a network in which a message can be broadcast in minimum possible (\(=\lceil \log_2 n \rceil\)) time from any vertex. Broadcast graphs which have the smallest number of edges are called \emph{Minimum Broadcast Graphs}, and are subjects of intensive study. In this paper, we study how the number of edges in minimum broadcast graphs decreases, as we allow additional time over \(\lceil \log_2 n \rceil\).

We improve results obtained by Shastri in [15] and prove a conjecture posed by Shastri in [15, 16].