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.