Given a connected graph \(G=(V,E)\) of order \(n\ge 2\) and two distinct vertices \(u,v\in V(G)\), consider two operations on \(G\): the \(k\)-multisubdivision and the \(k\)-path addition. Let \(msd_{\gamma_c}(G)\) and \(pa_{\gamma_c}(G)\) denote, respectively, the connected domination multisubdivision and path addition numbers of \(G\). In both operations, \(k\) represents the number of vertices added to \(V(G)\), resulting in a new graph denoted by \(G_{u,v,k}\). We prove that \(\gamma_c(G) \le \gamma_c(G_{u,v,k})\) for \(k = msd_{\gamma_c}(G) \in \{1,2,3\}\) in the case of \(k\)-multisubdivision, where \(uv \in E(G)\). Additionally, we show that \(\gamma_c(G) – 2 \le \gamma_c(G_{u,v,k})\) for \(k = pa_{\gamma_c}(G) \in \{0,1,2,3\}\) in the case of \(k\)-path addition, where \(uv \notin E(G)\), and provide both necessary and sufficient conditions under which these inequalities hold.

This paper introduces two novel sequences: the \(k-\)-division Fibonacci–Pell polynomials and the \(k-\)-division Gaussian Fibonacci–Pell polynomials. Building on the well-known Fibonacci and Pell sequences, these new sequences are defined using a division-based approach, enhancing their combinatorial and algebraic properties. We present explicit recurrence relations, generating functions, combinatorial identities, and Binet-type formulas for these sequences. A significant contribution of the study is the factorization of the Pascal matrix via the Riordan group method using the proposed polynomials. Two distinct factorizations are derived, highlighting the algebraic structure and combinatorial interpretations of the \(k-\)-division polynomials. The work not only generalizes known polynomial sequences but also provides new insights into their matrix representations and applications.

This paper provides new lower bounds for van der Waerden numbers using Rabung’s method, which colors based on the discrete logarithm modulo some prime. Through a distributed computing project with 500 volunteers over one year, we checked all primes up to 950 million, compared to 27 million in previous work. We point to evidence that the van der Waerden number for \(r\) colors and progression length \(k\) is roughly \(r^k\).

Richard Rado’s work in Ramsey Theory established conditions under which monochromatic solutions to a linear system must occur. In this paper, we find exact values for a linear system involving the equation \(x_1 + x_2 + c = x_0\) and two colors: Let \(c\) and \(k\) be integers with \(-1 \leq c \leq k\). Then the 2-color Rado number for the system \[\begin{array}{lcl} x_1 + x_2 + c &=& x_0, \\ y_1 + y_2 + k &=& y_0, \end{array}\] is infinite if \(c\) and \(k\) have opposite parity, and has a value of \(4k+5\) if \(c\) and \(k\) have the same parity. We also extend this to the continuous result where we color the real numbers.

In this article, we begin to investigate prime labelings of the zero-divisor graph of a commutative ring. A graph \(G\) with \(n\) vertices admits a prime labeling if the vertices can be labeled using distinct positive integers less than or equal to \(n\) such that any two adjacent vertices have labels which are relatively prime. We are able to construct several infinite families of commutative rings which will have prime labelings for their zero-divisor graphs. We also find infinite families of commutative rings which do not have prime labelings for their zero-divisor graphs. We then continue the process of determining which commutative rings will have prime labelings for their zero-divisor graphs by resolving the question for all rings with 14 or fewer vertices in their zero-divisor graph. We conclude with several unresolved questions that could be interesting to pursue further.

Some methods of constructions of square tactical decomposable regular group divisible designs are described. These designs are useful in threshold schemes. An L_2 design is also identified as square tactical decomposable. This completes spectrum of the solutions of entire L_2 designs listed in Clatworthy [2] using matrix approaches.

Lightweight design is widely used for optimizing machinery. This paper proposes an algebraic–topology–based structural optimization method that formulates a mathematical model for mechanical equipment. The model is solved via sequential linear programming and recast as a function-optimization problem addressed by a Memetic algorithm combining a Newtonian local search, adaptive multipoint crossover, and stochastic variability. Using a bridge crane main girder as a case study, we minimize cross-sectional area with selected sectional parameters as design variables and constraints from the Crane Design Manual. With population size 100 and 300 maximum iterations, the optimized girder achieves a 13% weight reduction. Static and dynamic analyses confirm that strength and stiffness satisfy safe working conditions.

Given a directed graph, the Minimum Feedback Arc Set (FAS) problem asks for a minimum (size) set of arcs in a directed graph, which, when removed, results in an acyclic graph. In a seminal paper, Berger and Shor [1], in 1990, developed initial upper bounds for the FAS problem in general directed graphs. Here we find asymptotic lower bounds for the FAS problem in a class of random, oriented, directed graphs derived from the Erdős-Rényi model \(G(n,M)\), with n vertices and M (undirected) edges, the latter randomly chosen. Each edge is then randomly given a direction to form our directed graph. We show that \[Pr\left(\textbf{Y}^* \le M \left( \frac{1}{2} -\sqrt{\frac{\log n}{\Delta_{av}}}\right)\right),\] approaches zero exponentially in \(n\), with \(\textbf{Y}^*\) the (random) size of the minimum feedback arc set and \(\Delta_{av}=2M/n\) the average vertex degree. Lower bounds for random tournaments, a special case, were obtained by Spencer [13] and de la Vega [3] and these are discussed. In comparing the bound above to averaged experimental FAS data on related random graphs developed by K. Hanauer [8] we find that the approximation \(\textbf{Y}^*_{av} \approx M\left( \frac{1}{2} -\frac{1}{2}\sqrt{\frac{\log n}{\Delta_{av}}}\right)\) lies remarkably close graphically to the algorithmically computed average size \(\textbf{Y}^*_{av}\) of minimum feedback arc sets.

An alternative proof of A.B. Evans’ result on the existence of strong complete mappings on finite abelian groups is presented. Applications of strong complete mappings in computing the chromatic numbers of~certain Cayley graphs are discussed.