A two-step approach to finding knight covers for an \(N \times N\) chessboard eliminates the problem of detecting duplicate partial solutions. The time and storage needed to generate solutions is greatly reduced. The method can handle boards as large as \(45 \times 45\) and has matched or beaten all previously known solutions for every board size tried.

In this paper we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-1}{2} \leq m \leq \frac{n^2-n}{2}\), when \(n\) is odd. Moreover, when \(n\) is even we prove that there exists a strong critical set of size \(m\) in the back circulant latin square of order \(n\) for all \(\frac{n^2-n}{2}-(n-2) \leq m \leq \frac{n^2-n}{2}\) and \(m \in \{\frac{n^2}{4}, \frac{n^2}{4}+2, \frac{n^2}{4}+4, \ldots, \frac{n^2-n}{2}-n\}\).

In this paper, a characterization of two classes of \((q, q+1)\)-geometries, that are fully embedded in a projective space \(PG(n, q)\), is obtained. The first class is the one of the \((q,q+1)\)-geometry \(H^{n,m}_q\), having points the points of \(PG(n, q)\) that are not contained in an \(m\)-dimensional subspace \(\Pi[m]\) of \(PG(n, q)\), for \(0 \leq m \leq n-3\), and lines the lines of \(PG(n, q)\) skew to \(\Pi[m]\). The second class is the one of the \((q,q+1)\)-geometry \(SH^{n,m}_q\), having the same point set as \(H^{n,m}_q\), but with \(-1 \leq m \leq n-3\), and lines the lines skew to \(\Pi^{n,m}_q\) that are not contained in a certain partition of the point set of \(SH^{n,m}_q\). Our characterization uses the axiom of Pasch, which is also known as axiom of Veblen-Young. It is a generalization of the characterization for partial geometries satisfying the axiom of Pasch by J. A. Thas and F. De Clerck. A characterization for \(H^{n,m}_q\) was already proved by H. Cuypers. His result however does not include \(SH^{n,m}_q\).

In this note we construct nested partially balanced incomplete block designs based on \(NC_{m}\)-scheme. Secondly we construct NPBIB designs from a given PBIB design with \(\lambda_{1} = 1\) and \(\lambda_{2} = 0\) with same association scheme for both systems of PBIB designs. Finally, we give some results and examples where the two systems of PBIB designs in NPBIB designs have different association schemes.

This paper discusses the covering property and the Uniqueness Property of Minima (UPM) for linear forms in an arbitrary number of variables, with emphasis on the case of three variables (triple loop graph). It also studies the diameter of some families of undirected chordal ring graphs. We focus upon maximizing the number of vertices in the graph for given diameter and degree. We study the result in \([2]\), we find that the family of triple loop graphs of the form \(G(4k^2+2k+1; 1;2k+1; 2k^2)\) has a larger number of nodes for diameter \(k\) than the family \(G(3k^2+3k+1;1;3k+1;3k+2)\) given in \([2]\). Moreover, we show that both families have the Uniqueness Property of Minima.

In this paper, an algorithm based on. trades is presented to classify two classes of large sets of \(t\)-designs, namely \(LS[14](2, 5, 10)\) and \(LS[6](3, 5, 12)\).

In this work, we study which tubular surfaces verify that the embeddings of infinite, locally finite connected graphs without vertex accumulation points are embeddings without edge accumulation points. Furthermore, we characterize the graphs which admit embeddings with no edge accumulation points in the sphere with \(n\) ends in terms of forbidden subgraphs.

In this paper, self-centered, bi-eccentric splitting graphs are characterized. Further various bounds for domination number, global domination number and the neighborhood number of these graphs are obtained.

In this study we are going to give a new \((t,k)\)-geodetic set definition. This is a refinement of the geodetic set definition given in \([11]\). With this new definition we obtain more information about the graph. We also give a relationship between the \((t,k)\)-geodetic set and the integrity of a graph. By using a \((t,k)\)-geodetic set we give a new proof for the upper bound of integrity of trees and unicycle graphs.

For a long time we had thought that there does not exist an OGDD of type \(4^4\). In this article, an OGDD of type \(4^4\) will be constructed.