Recently, it was shown that the Gallai-Ramsey number satisfies \(gr(F_{3,2}, K_3, K_3)=31\), where \(F_{3,2}\) is the generalized fan \(F_{3,2}:=K_1+2K_3\). In this paper, we show that the star-critical Gallai-Ramsey number satisfies \(gr_*(F_{3,2}, K_3, K_3)=27\). We also prove that the critical colorings for \(r_*(K_3, K_3)\), \(gr(F_{3,2},K_3,K_3)\), and \(gr_*(F_{3,2},K_3,K_3)\) are unique.

Given a network modeled as a graph, a detection system is a subset of vertices equipped with “detectors” that can uniquely identify an “intruder” anywhere in the graph. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). In an OLD set, each vertex has a unique, non-empty set of detectors in its open neighborhood; meanwhile, in an IC, each vertex has a unique, non-empty set of detectors in its closed neighborhood. We explore one of their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector retains the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king grid, and presents the proof for the bounds on their minimum densities; \(\left[\frac{3}{10}, \frac{1}{3}\right]\) for RED:OLD sets and \(\left[\frac{3}{11}, \frac{1}{3}\right]\) for RED:ICs.

Exploring the vulnerability of any real-life network helps designers understand how strongly components or elements of the network are connected and how well they can function if there is any disruption. Any chemical structure can also be considered as a network in which the atoms correspond to the vertices, and the chemical bonds between the atoms correspond to the edges. Let \(G=(V, E)\) represent any simple graph with vertex set \(V\) and edge set \(E\). The vulnerability measure used in this paper is the paired domination integrity, defined as the minimum of the sum of any paired dominating set \(S\) of a graph \(G\) and the order of the largest component in the induced subgraph of \(V-S\). The minimum is found by considering all possible paired dominating sets of \(G\). In this paper, we obtain the paired domination integrity of the comb product of paths and cycles. In addition, we extend the study of graph vulnerability to chemical structures.

Let \(k, b, n\) be positive integers such that \(b\geq 2\). Denote by \(S(k,b,n)\) the numerical semigroup generated by \(\left\{b^{k+n+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\right\}\). In this paper, we give formulas for computing the embedding dimension and the Frobenius number of \(S(k,b,n)\).

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 \(r^k\).