A \(2\)-factored dominating set (\(2\)fd-set) of a graph \(G=(V,E)\) is a dominating set \(F\subseteq V\) such that the induced subgraph \(G[F]\) is \(2\)-regular, and hence is a disjoint union of cycles. In this study, \(2\)-factored dominating sets on fixed-width grid graphs of dimensions \(m \times n\), where \(m \in \{2,3,4\}\), are enumerated. We establish theorems describing the generating functions with respect to the number of \(2\)-factored dominating sets in these grid graphs. The number of \(2\)-factored dominating sets grows exponentially with \(n\), with growth constant determined by the dominant singularity of the generating function.

A Boolean network maps Boolean configurations of fixed length to themselves. A trapspace is an invariant subcube; a principal trapspace is the smallest trapspace containing a configuration, while a minimal trapspace contains no proper trapspace. Commutative Boolean networks are those whose local updates commute. We connect these concepts through five contributions. First, we introduce trapping graphs and trapping closures, define trapping networks by transitivity of their general asynchronous graphs, and prove that they are exactly the trapping closures. Second, we show that two Boolean networks have the same collections of principal trapspaces if and only if they have the same trapping closure. Hence, trapping networks provide a normal form for trapspace analysis. We also characterize the possible collections of principal and minimal trapspaces. Third, we prove that commutative networks are trapping and classify their principal trapspaces. Fourth, we study bijective commutative networks, called Marseille networks, and give equivalent characterizations and classifications. Fifth, we study idempotent commutative networks, called Lille networks, relate them to globally idempotent networks, prove that globally idempotent networks are trapping, and provide equivalent characterizations. These results clarify the relationships among asynchronous, general asynchronous, and trapping graphs and describe the structure of trapping networks.

The Ramsey number \(R(G)\) of a graph \(G\) without isolated vertices is the minimum positive integer \(n\) such that for every red-blue coloring of the complete graph \(K_n\) of order \(n\), there is a subgraph isomorphic to \(G\) all of whose edges are colored the same (a monochromatic \(G\)). A Ramsey chain in a graph \(G\) with a red-blue coloring is a sequence \(G_1\), \(G_2\), \(\ldots\), \(G_{k}\) of pairwise edge-disjoint monochromatic subgraphs of \(G\) such that \(G_i\) has \(i\) edges for \(1 \le i \le k\) and \(G_i\) is isomorphic to a subgraph of \(G_{i+1}\) for \(1 \le i \le k-1\). The subgraphs in a Ramsey chain are the links of the chain and the terminal subgraph \(G_k\) is the target link of the chain. A graph \(H\) without isolated vertices is called a target graph if there exists a positive integer \(n\) such that every red-blue coloring of \(K_n\) results in a Ramsey chain with target link \(H\). The target Ramsey number \(TR(H)\) of \(H\) is the minimum positive integer \(n\) such that every red-blue coloring of \(K_n\) results in a Ramsey chain with target link \(H\). The target Ramsey number \(TR(s)\) of a Ramsey chain \(s\) is the minimum positive integer \(n\) such that \(s\) is a Ramsey chain in every red-blue coloring of \(K_{n}\). We investigate graphs \(H\) with the property that \(TR(s) =TR(H)= R(H)\) for every Ramsey chain \(s\) with target link \(H\). It is shown that every graph \(H\) with relatively small size has this property. Other results and open questions are also presented.

The sum graph \(G^{+}(S)\) of a finite subset \(S\subset \mathbb{N}\)={1,2,3,…} is the graph (V,E) where V=S and \(uv\in\)E if and only if \(u+v\in\) S. This concept was introduced by Harary [9], where some basic properties of the family of all sum graphs were presented. In [10], Harary extended this definition into an integral sum graph and proposed some open problems. Motivated by these definitions, we introduce a graph called perfect difference graph. We investigate the properties of this family of graphs.

We study the Equivalent Local Sequence Problem (ELSP) for simple undirected graphs using Bouchet’s isotropic-system formalism and normal matrices over \(\mathbb F_2\). Although Bouchet’s theory characterizes graph local equivalence in polynomial time, converting normal-matrix certificates into explicit graph transformations remains challenging. We introduce the Normal-Matrix Factorization Problem (NMFP), which asks whether a normal-matrix witness of local equivalence has a graph-compatible factorization into elementary transformations. Whenever such a factorization exists, an explicit local-complementation sequence can be recovered in polynomial time. Thus, the constructive part of ELSP reduces to NMFP, identifying normal-matrix factorization as its main unresolved algebraic difficulty. We apply this framework to undirected Paley graphs. In contrast to the directed case, whose local-complementation dynamics are abelian and admit linear inversion, the undirected case is noncommutative and has a more intricate stabilizer structure. Using normal matrices, we analyze Paley-graph orbits and stabilizers, derive algebraic constraints on stabilizing transformations, and completely verify the first undirected Paley graph \(P_5\). These results establish NMFP as central to constructive local equivalence and reveal connections among isotropic systems, graph transformations, and algebraic stabilizers.

We relate two decompositions that isolate König–Egerváry structure: the SD–KE decomposition, defined by the classical Edmonds–Sterboul–Deming obstructions, and Larson’s critical independence decomposition, whose complement is \(2\)-bicritical. Let \(I\) be a maximum critical independent set, set \(L(G)=I\cup N_G(I)\), \(L^c(G)=V(G)\setminus L(G)\), and \(B=N_G(I)\). We first prove a normal form for maximum matchings: each maximum matching is obtained from a maximum matching of \(G[L^c(G)]\), a matching crossing from a subset \(X\subseteq B\) into exposed vertices on the \(L^c(G)\) side, and a maximum matching of \(G[L(G)-X]\). The possible sets \(X\) are the crossing profiles. For a crossing profile \(X\) and a maximum matching \(P\) of \(G[L(G)-X]\), view \(P\) as a matching of \(G[L(G)]\). Let \(\Lambda_I(G)\) be the union of the vertices on \(P\)-augmenting paths in \(G[L(G)]\), over all crossing profiles and all residual maximum matchings. We prove \(SD(G)=L^c(G)\cup\Lambda_I(G),\) and \(KE(G)=L(G)\setminus\Lambda_I(G).\) Thus the SD vertices on the Larson König–Egerváry side are precisely those reached by augmenting corridors forced by admissible crossings.

An edge labeling of a graph \(G = (V, E)\) is said to be local antimagic if it is a bijection \(f:E \to\{1,\ldots ,|E|\}\) such that for any pair of adjacent vertices \(x\) and \(y\), \(f^+(x)\not= f^+(y)\), where the induced vertex label \(f^+(x)= \sum f(e)\), with \(e\) ranging over all the edges incident to \(x\). The local antimagic chromatic number of \(G\), denoted by \(\chi_{la}(G)\), is the minimum number of distinct induced vertex labels over all local antimagic labelings of \(G\). In this paper, we study local antimagic labeling of three disjoint cycles with at least two odd cycles, one of which is \(C_3\). We prove that (1) \(\chi_{la}(C_3+C_3+C_3)=5\) and \(\chi_{la}(C_3+C_3+C_{k})=4\) for \(k\geq 4\); (2) \(\chi_{la}(C_3+C_4+C_{2k+1})=4\); (3) \(\chi_{la}(C_3+C_6+C_{4k+1})=3\); (4) \(\chi_{la}(C_3+C_{10}+C_{8k-3})=3\); (5) \(\chi_{la}(C_3+C_5+C_{14})=4\) and \(\chi_{la}(C_3+C_5+C_{4k+2})=3\) for \(k\neq3\); (6) \(\chi_{la}(C_3+C_{4k+1}+C_{4k+5})=3\) and \(\chi_{la}(C_3+C_{4k+1}+C_{4k+6})=3\).

This paper identifies and corrects invalid arguments in [Edge \(k\)-Product Cordial Labeling of Graphs, Eur. J. Pure Appl. Math., 18(2), Article No. 5887 (2025)]. In particular, several published statements are shown to be false through explicit counterexamples. Revised versions of these results are established under appropriate conditions, often involving structural properties such as girth. Correct bounds or conditions are obtained for specific graph classes, including shadow graphs, splitting graphs of stars, and path unions of cycles. These corrections clarify the limitations of the earlier claims and provide a more accurate foundation for further study of edge \(k\)-product cordial labeling.

This work introduces Combinatorial Grid Sequencing (CGS), an axiomatic combinatorial theory inspired by certain patterns of concrete-casting works in multi-storey building construction. By incorporating elementary number theory and max-plus algebra, we derive several theorems that characterize the inherent properties of CGS systems. Notably, the development of an explicit inverse map within this framework led to the independent discovery of three number-theoretic identities involving the floor function. The theory provides a rigorous framework for developing algorithms to model real-world problems categorized as CGS problems. We demonstrate that this approach offers a time-efficient, systematic computational model capable of significantly accelerating certain complex optimization tasks for engineers.

It is known that an automorphism group \(G\) acting on a symmetric, uniform, and balanced incidence structure \({\cal D}\) has the same number of orbits on the set of points and the set of lines of \({\cal D}\). Moreover, a group \(G\) induces a tactical decomposition of \({\cal D}\). These facts are often used to perform efficient constructions of various combinatorial designs and other incidence structures. In this paper, we use a variation of this approach to construct Hadamard 3-balanced incidence structures by employing an automorphism of order 3. We are also able to reach structures with a small automorphism group.

A signed Roman dominating function (SRDF) on \(G=(V,E)\), a finite, connected, simple graph is a mapping \(f : V \to \{-1, 1, 2\},\) such that

(a) For every vertex \(x \in V\), \(\sum\limits_{y \in N[x]} f(y) \ge 1,\) where \(N[x]\) denotes the closed neighborhood of \(x\), consisting of \(x\) together with all vertices adjacent to \(x\).

(b) Every vertex \(x \in V\) with \(f(x) = -1\) is adjacent to at least one vertex \(y \in V\) such that \(f(y) = 2\).

The weight of an SRDF is defined as \(\sum\limits_{v \in V(G)} f(v)\). The signed Roman domination number (SRDN) of \(G\), denoted by \(\gamma_{SR}(G)\), is the minimum possible weight among all signed Roman dominating functions on \(G\). In this work, we determine the signed Roman domination number of the ladder graph \(LG_n\) and its complement \(LG_n^c\).

The triplication method for constructing strong starters in \(\mathbb{Z}_{3m}\) from starters in \(\mathbb{Z}_{m}\) (say, a starter of order 21 from a starter of order 7) was proposed by the authors in 2025. The method reduced the construction of this particular combinatorial design (a strong starter in a cyclic group) to solving a Sudoku-type problem – an independent task with its own tools and techniques available. The Sudoku-type problem was formulated in terms of the so-called triplication table constructed from a starter of order \(m\). The method was applicable to odd orders \(m\ge 7\) not divisible by 3. In the present paper, our previous approach is developed in two directions: (1) the definition of the triplication table is generalized, which expands possibilities for its construction to include three base starters, “pseudostarters”, or even more general setup; (2) the formulation of the Sudoku-type problem is broadened to embrace various scenarios of “modular encoding” and reconstruction of strong starters from its solution. A theoretical gain of these developments is an improved understanding of the general structure of the triplication approach. A practical outcome is that all odd values \(m \ge 5\) (including those divisible by 3) are now admissible and the set of possible triplication tables is so broad that any latent strong starter of odd order \(3m\) can emerge by triplication.

For the prism graphs \(G_n=C_n\square P_2\), the chromatic polynomial has an explicit four-branch transfer-matrix expansion with polynomial eigenvalues and amplitudes. The Beraha-Kahane-Weiss (BKW) theorem then confines asymptotic root accumulation to equimodular ties and amplitude zeros. Here the only amplitude zeros are \(z=1\) and \(z=\frac{3\pm\sqrt5}{2}\), and a dominance check shows that none yields an isolated BKW limit point. A complete algebraic classification of the prism tie curves appears in [1]. The dominant quadratic-linear ties are recast here in a centered Cassini-type normal form, giving a product-of-distances interpretation together with explicit quartic implicit and centered polar equations. A global Rouché comparison also yields a uniform finite-\(n\) bound: every chromatic root of \(G_n\) satisfies \(|z|<6\) for all \(n\ge3\).

For a fixed integer \(k\geq0\), let \(p_k(n)\) denote the number of integer partitions \(\lambda=(\lambda_1,\ldots,\lambda_r)\) of \(n\) satisfying the minimal-difference condition \(\lambda_i-\lambda_{i+1}\geq k,\quad 1\leq i<r,\) with the convention that the smallest part is at least one. We study the logarithmic asymptotic growth of \(p_k(n)\) through the length-refined generating function \(P_k(q)=\sum_{r\geq0}\frac{q^{\,r+k\binom r2}}{(q;q)_r}.\) The factor \(q^r\) is essential and comes from the condition that each part is positive. For \(k\geq1\), we prove that \(\log p_k(n)\sim B_k\sqrt n,\) where \(B_k=2\sqrt{A_k}\) and \(A_k=\frac{\pi^2}{6}-Li_2(e^{-x_k})-\frac{k}{2}x_k^2,\) with \(x_k>0\) the unique solution of \(1-e^{-x_k}=e^{-kx_k}\). The case \(k=0\) is stated separately and gives the classical Hardy–Ramanujan constant \(B_0=\pi\sqrt{2/3}\). The proof combines a uniform Euler–Maclaurin estimate for the truncated Euler product, a discrete Laplace principle for the length sum, and Ingham’s Tauberian theorem.

The enumeration of noncrossing trees has attracted significant attention since the turn of 21st century. These trees have been studied with respect to various statistics, including the number of vertices, leaves, root degree, and levels. In contrast, plane trees have a longer history of exploration. In 2010, Deutsch and his co-authors introduced and enumerated a class of plane trees in which a rightmost edge may be marked, provided it does not lead to a leaf. Their enumeration formula involved the Catalan numbers, which also count plane trees. In this work, we extend the concept of marking rightmost edges to noncrossing trees, introducing a new combinatorial structure. We enumerate this structure according to the number of edges, marked edges, root degree, and leaves. We use symbolic method and Lagrange Inversion Formula to derive our results. Furthermore, we establish connections between these new structures and both labelled plane trees and ternary trees.

By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an \(n\)-column board after \(k\) color-alternating moves. For fixed \(k\) we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as \(n \to \infty\). We then turn our attention to the fixed-\(n\) case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as \(k \to \infty\).

Sterboul’s theorem characterizes non-Kőnig–Egerváry graphs by the presence, relative to a maximum matching, of a flower or a posy. In this paper we translate that obstruction into the language of perfect flowers and the core of the graph. We introduce core-defective perfect flowers: perfect flowers whose alternating path contains a vertex at odd distance from the blossom base that does not belong to the core. We prove first that every Kőnig–Egerváry graph is core-rigid: in every perfect flower, all odd-distance vertices of the attaching path lie in the core. Conversely, if \(G\) is connected and is not an odd cycle, then \(G\) is non-Kőnig–Egerváry if and only if \(G\) contains a core-defective perfect flower. Thus, among connected graphs different from an odd cycle, the Kőnig–Egerváry graphs are exactly the graphs with no core-defective perfect flower. In the matchable case the statement strengthens: if \(G\) has a perfect matching, then being non-Kőnig–Egerváry is equivalent to the existence of a core-defective perfect flower for some maximum matching, and also equivalent to the existence of one for every maximum matching. We include examples and counterexamples showing why odd cycles, disconnected graphs, and the universal quantifier over maximum matchings require separate treatment.

A hypergraph \(H\) is said to be \(r\)-partite \(r\)-uniform if its vertex set \(V\) can be partitioned into non-empty sets \(V_1, V_2, \cdots, V_r\) so that every edge in the edge set \(E(H)\), consists of precisely one vertex from each set \(V_i\), \(i=1,2,\cdots,r\). It is denoted as \(H^r(V_1,V_2,\cdots,V_r)\) or \(H^r_{(n_1,n_2,\cdots,n_r)}\) if \(|V_i|=n_i\) for \(i=1,2,\cdots,r\). There exists an \(r\)-partite self-complementary \(r\)-uniform hypergraph \(H^r(V_1,V_2,\cdots,V_r)\) where \(|V_i|=n_i\) for \(i=1,2,\cdots,r\) if and only if at least one of \(n_1,n_2,\cdots,n_r\) is even. And there exists an \(r\)-partite almost self-complementary \(r\)-uniform hypergraph \(H^r(V_1, V_2,\cdots,V_r)\) where \(|V_i|=n_i\) for \(i=1,2,\cdots,r\) if and only if \(n_1,n_2,\cdots,n_r\) are odd. In this paper, we prove the existence of regular \(3\)-partite self-complementary \(3\)-uniform hypergraphs. Further we prove there does not exist a regular \(3\)-partite almost self-complementary \(3\)-uniform hypergraph.

We study the difference between the numbers of even and odd permutations in \(\mathfrak{S}_n\) having exactly \(k\) fixed points. We derive a closed formula for this quantity using four complementary approaches: exponential generating functions, a determinant representation, a combinatorial derivation based on inclusion–exclusion on cycle structures, and a factorization via the stabilizer subgroup, through restriction to the complement of the fixed-point set. The resulting expression provides a signed refinement of the classical rencontres numbers and yields a simple polynomial form for the associated signed fixed-point distribution.