We propose and study the problem of finding the smallest nonnegative integer \(s\) such that a GDD\((m, n, 3; 0, \lambda)\) can be embedded into a BIBD\((mn + s, 3, \lambda)\). We find the values of \(s\) for all cases except for the case where \(n \equiv 5 \pmod{6}\) and \(m \equiv 1, 3 \pmod{6}\) and \(m \ge 3\), which remains as an open problem.

A simple graph \(G\) with \(p\) vertices is said to be vertex-Euclidean if there exists a bijection \(f: V(G) \rightarrow \{1, 2, \ldots, p\}\) such that \(f(v_1) + f(v_2) > f(v_3)\) for each \(C_3\)-subgraph with vertex set \(\{v_1, v_2, v_3\}\), where \(f(v_1) < f(v_2) < f(v_3)\). More generally, the vertex-Euclidean deficiency of a graph \(G\) is the smallest integer \(k\) such that \(G \cup N_k\) is vertex-Euclidean. To illustrate the idea behind this new graph labeling problem, we study the vertex-Euclidean deficiency of two new families of graphs called the complete fan graphs and the complete wheel graphs. We also explore some related problems, and pose several research topics for further study.

A signed magic rectangle \(SMR(m, n; r, s)\) is an \(m \times n\) array with entries from \(X\), where \(X = \{0, \pm1, \pm2, \ldots, \pm(ms – 1)/2\}\) if \(mr\) is odd and \(X = \{\pm1, \pm2, \ldots, \pm mr/2\}\) if \(mr\) is even, such that precisely \(r\) cells in every row and \(s\) cells in every column are filled, every integer from set \(X\) appears exactly once in the array, and the sum of each row and of each column is zero. In this paper, we prove that a signed magic rectangle \(SMR(m, n; r, 2)\) exists if and only if \(m = 2\) and \(n = r \equiv 0, 3 \pmod{4}\) or \(m, r \geq 3\) and \(mr = 2n\).

A graph \(G\) is asymmetric if its automorphism group is trivial. Asymmetric graphs were introduced by Erdős and Rényi [1]. They suggested the problem of starting with an asymmetric graph and removing some number, \(r\), of edges and/or adding some number, \(s\), of edges so that the resulting graph is non-asymmetric. Erdős and Rényi defined the degree of asymmetry of a graph to be the minimum value of \(r+s\). In this paper, we consider another property that measures how close a given non-asymmetric graph is to being asymmetric. Brewer et al defined the asymmetric index of a graph \(G\), denoted \(ai(G)\) is the minimum of \(r+s\) so that the resulting graph \(G\) is asymmetric [2]. It is noted that \(ai(G)\) is only defined for graphs with at least six vertices. We investigate the asymmetric index of both connected and disconnected graphs including paths, cycles, and grids, with the addition of up to two isolated vertices. Furthermore for a graph in these families \(G\) we determine the number of labelled asymmetric graphs that can be obtained by adding or removing \(ai(G)\) edges. This leads to the related question: Given a graph \(G\) where \(ai(G)=1\), what is the probability that for a randomly chosen edge \(e\), that \(G-e\) will be asymmetric? A graph is called minimally non-asymmetric if this probability is \(1\). We give a construction of infinite families of minimally non-asymmetric graphs.

A graph \(G\) is said to arrow the graphs \(F\) and \(H\), written \(G \rightarrow (F, H)\), if every red-blue coloring of \(G\) results in a red \(F\) or a blue \(H\). The primary question has been determining graphs \(G\) for which \(G \rightarrow (F, H)\). If we consider the version for which \(F = H\), then we can ask a different question: Given a graph \(G\), can we determine all graphs \(F\) such that \(G \rightarrow (F, F)\), or simply \(G \rightarrow F\)? We call this set of graphs the down-arrow Ramsey set of \(G\), or \(\downarrow G\). The down-arrow Ramsey set of cycles, paths, and small complete graphs are determined. Graph ideals and graph intersections are introduced and a method for computing down-arrow Ramsey sets is described.

We use a dynamic programming algorithm to establish a new lower bound on the domination number of complete cylindrical grid graphs of the form \(C_n\square P_m\), that is, the Cartesian product of a path and a cycle, when \(n\equiv 2\pmod{5}\), and we establish a new upper bound equal to the lower bound, thus computing the exact domination number for these graphs.

In this paper, we address computational questions surrounding the enumeration of non-isomorphic André planes for any prime power order \(q\). We are particularly focused on providing a complete enumeration of all such planes for relatively small orders (up to 125), as well as developing computationally efficient ways to count the number of isomorphism classes for other orders where enumeration is infeasible. André planes of all dimensions over their kernel are considered.

We use a dynamic programming algorithm to establish a lower bound on the domination number of complete grid graphs of the form \(C_n\square P_m\), that is, the Cartesian product of a cycle \(C_n\) and a path \(P_m\), for \(m\) and \(n\) sufficiently large.

With the increasingly frequent exchanges between countries, my country’s demand for high-quality English translators has greatly increased. However, an important problem we are currently facing is that China’s translation talents are far behind the demand. An important reason for this phenomenon is that the traditional translation teaching is difficult to cultivate translators who can meet the market demand. Therefore, it is necessary to improve the traditional translation teaching mode. Translation teaching for English majors is an important part of translation teaching. Therefore, after evaluating the speech characteristics and speech data, this document first proposes a translation classification error detection model based on mfcc-rf. The acoustic function of the extracted 39 dimensional Mel inverse spectral coefficient is the input of the random forest classifier, and a classification error detection model is established. By analyzing the experimental results, the MFCC radio frequency translation error detection model has achieved high classification error detection accuracy under three types of errors (rising, falling and shortening). The experimental results show that, with semantic similarity as the design principle of distractors, using the word vector training method of the context word prediction model to automatically generate distractors can ultimately improve the comprehensive training efficiency of college English majors’ translation ability.

A node in the \(n\)-dimensional hypercube \(Q_n\) is called an odd node (resp. an even node) if the sum of all digits of the node is odd (resp. even). Let \(F\subset E(Q_n)\) and let \(L\) be a linear forest in \(Q_n-F\) such that \(|E(L)|+|F|\leq n-2\) for \(n\geq 2\). Let \(x\) be an odd node and \(y\) an even node in \(Q_n\) such that none of the paths in \(L\) has \(x\) or \(y\) as internal node or both of them as end nodes. In this note, we prove that there is a Hamiltonian path between \(x\) and \(y\) passing through \(L\) in \(Q_n-F\). The upper bound \(n-2\) on \(|E(L)|+|F|\) is sharp.