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 \(r^k\).

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.

Let \(G = (V, E)\) be a simple graph. A subset \(D \subseteq V\) is called a dominating set of \(G\) if every vertex in \(V\) is either in \(D\) or has a neighbour in \(D\). A subset \(D \subseteq V\) is called an equitable dominating set of \(G\) if for every vertex \(v \in V \setminus D\), there exists a vertex \(u \in D\) such that \(uv \in E(G)\) and \(\lvert d_G(u) – d_G(v) \rvert \leq 1\). The minimum cardinality of an equitable dominating set of \(G\), denoted by \(\gamma^{e}(G)\), is called the equitable domination number of \(G\). In this paper, we study the equitable domination number of certain graph operators such as the double graph, the Mycielskian, and the subdivision of a graph.

This paper studies the Pell-Narayana sequence modulo \(m\). It starts by defining the Pell-Narayana numbers and examining their combinatorial relationships with well-known sequences and functions, including Eulerian, Catalan, and Delannoy numbers. Building on this, the concept of a Pell-Narayana orbit is introduced for a 2-generator group with generating pair \((x, y) \in G\), which allows the analysis of the periods of these orbits. The results include explicit calculations of the Pell-Narayana periods for polyhedral and binary polyhedral groups, depending on the choice of generating pair \((x, y)\), along with a discussion of their properties. Furthermore, the paper determines the periodic lengths of Pell-Narayana orbits for the groups \(Q_8, Q_8 \times \mathbb{Z}_{2m},\) and \(Q_8 \times_\varphi \mathbb{Z}_{2m}\) for all \(m \geq 3\).

A graph \(G=(V,E)\) is \(H\)-supermagic if there exists a bijection \(f\) from the set \(V\cup E\) to the set of integers \(\{1,2,3,\dots,|V|+|E|\}\), called \(H\)-supermagic labeling such that the sum of labels of all elements of every induced subgraph of \(G\) isomorphic to \(H\) is equal to the same integer and all vertex labels are in \(\{1,2,3,\dots,|V|\}\). We present a \((K_4-e)\)-supermagic labeling of the triangular ladder \(TL_{2n}\) for any \(n\geq2\).

Slilaty and Zaslavsky (2024) characterized all single-element extensions of graphic matroids in terms of a graphical structure called a cobiased graph. In this paper we characterize all orientations of a single-element extension of a graphic matroid in terms of graphically defined orientations of its associated cobiased graph. We also explain how these orientations can be canonically understood as dual to orientations of biased graphs if and only if the underlying graph is planar.

We consider a combinatorial question about searching for an unknown ideal \(\mu\) within a known pointed poset \(\lambda\). Elements of \(\lambda\) may be queried for membership in \(\mu\), but at most \(k\) positive queries are permitted. We provide a general search strategy for this problem, and establish new bounds (based on \(k\) and the degree and height of \(\lambda\)) for the total number of queries required to identify \(\mu\). We show that this strategy performs asymptotically optimally on the family of complete \(\ell\)-ary trees as the height grows.