A new series of four-associate class partially balanced incomplete block designs in two replications has been proposed. The blocks of these designs are of two different sizes. The blocks can be divided into two groups such that every treatment appears in each group exactly once, and any two blocks belonging to two different groups have a constant number of treatments in common, i.e., these designs are affine resolvable.

Let \( 0<k\in\mathbb{Z} \). Let the star 2-set transposition graph \( ST^2_k \) be the \( (2k-1) \)-regular graph whose vertices are the \( 2k \)-strings on \( k \) symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such \( 2k \)-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph \( PC^2_k \) has the same vertex set of \( ST^2_k \) and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of \( ST^2_k \). For \( 1<k\in\mathbb{Z} \), we show that \( ST^2_k \) and \( PC^2_k \), among other intermediate transposition graphs, have total colorings via \( 2k-1 \) colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of \( ST^2_k \) and \( PC^2_k \), and partitions into \( 2k-1 \) such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

The scientific knowledge graph is an emerging research method in this context. In the research of physical education teaching, the research and sorting out of the research results of physical education teaching in my country from the perspective of scientometrics and information visualization is still slightly insufficient. The similarity between the frontiers of physical education teaching research in China and the United States in the past five years is that both countries have paid more attention to research topics such as physical education teaching methods and physical education courses. This paper proposes a rough set knowledge reduction algorithm based on improved genetic algorithm. The support and importance of conditional attributes to decision attributes are introduced into the information system, which are added to the genetic algorithm as heuristic information, and the concepts of population dissimilarity and individual dissimilarity are proposed to improve the genetic algorithm. The research on school physical education in my country is biased towards problem research, while the research on physical education teaching methods in the United States is biased towards student health; In addition, starting from the national conditions, the hotspots in the field of physical education teaching in my country tend to be “Sports and Health Curriculum Standards”, physical education teachers, physical education ideas, educational theories and college sports, while the hotspots in the field of physical education teaching in the United States tend to be physical activity, children and adolescents , students, women, exercise education, physical education, self-determination theory, and the integration of psychological motivation and physical education. Experimental data analysis my country’s physical education curriculum research should appropriately increase the attention to the details of physical education curriculum, and my country’s physical education teaching practice research should appropriately increase the research on physical education from the perspective of public health.

In secret sharing, the relationships between participants and the information they hold can be modeled effectively using graph structures. Graphs allow us to visualize and analyze these relationships, making it easier to define access structures, optimize share distributions, and ensure security. This paper provides the first comprehensive review of existing research on the application of graph theory to secret sharing comparing different classic and modern approaches and analyzing the current litterature. Through this study we highlight the key advances and methodologies that have been developed, underscoring the pivotal role of graph theoretic approaches in enhancing the security and efficiency of secret sharing schemes. Furthermore, the review identifies open challenges and future research directions, providing insights into potential innovations that could further strengthen cryptographic practices. This work serves as a foundational resource for researchers and practitioners seeking to deepen their understanding of the intersection between graph theory and secret sharing, fostering the development of more robust and sophisticated cryptographic solutions.

This paper investigates the Turan-like problem for \(\mathcal{K}^-_{r + 1}\)-free \((r \geq 2)\) unbalanced signed graphs, where \(\mathcal{K}^-_{r + 1}\) is the set of unbalanced signed complete graphs with \(r+1\) vertices. The maximum number of edges and the maximum index for \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs are given. Moreover, the extremal \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs with the maximum index are characterized.

In this paper, we give a classification of all Mengerian \(4\)-uniform hypergraphs derived from graphs.

Big data technology makes it possible to scientifically analyse a large amount of marketing data, which plays an important role in the development of marketing strategies for products and the improvement of marketing effects. In this paper, a marketing data stream analysis system is designed based on the stream analysis method. The system designs and optimises the marketing data storage and retrieval, data acquisition and streaming calculation engine to achieve real-time user behaviour data streaming analysis. The average response time accuracy of the system’s data can reach 96%, the throughput rate is 11.8% ahead of the maximum compared to the Word Count system, and the before-and-after ratios of the PUSH message click rate, the user registration success rate, the online shop attention rate, the returning customer rate, and the loyal customer rate are 1.03, 1.02, 1.27, 1.11, 1.27, and 1.78, respectively. It indicates that this paper’s design of the marketing data streaming analysis system has good performance and application effect.