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.
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.
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.
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.
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 \(n\ge3\).
