组合数学

We study truncation compatible families F = (F_m)_{m>=1} over Q[z] through an inverse limit formalism, and we evaluate them at the punctured cyclotomic cosine points alpha_{k,n} = cos(2 pi k/n) with the specialization z equals n-1. For…

Let $L(G)$ be the set of all subgroups of a group $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b)…

The Tur\'{a}n number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Tur\'an problems of a…

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In…

For a specific rooted labeled tree topology, a labeled history is a sequence of branchings that give rise to that labeled topology as it unfolds over time. Here, for $r$-furcating trees, we use a connection with Huffman trees from…

A graph $G$ is called matching covered if all of its edges are contained in some perfect matching of $G$. Furthermore, a cycle $C \subseteq G$ is called conformal if $G - V(C)$ has a perfect matching and $G$ itself is called cycle-conformal…

A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in…

Lights Out is a game played on a graph $G$ where every vertex has a light bulb that is either on or off, and pressing a vertex $v$ toggles the state of every vertex in the closed neighborhood of $v$. The goal is to find a subset of vertices…

The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $(claw, diamond, K_4)$-free graphs. Recently, $3$-colorability has been also considered for $(claw, N_{1,1,1})$-free graphs. Here, a generalised…

This article examines matrices whose entries are determined by recursive relations of the form $A_{i, j} = x A_{i, j-1} + y A_{i-1, j-1} + z A_{i-1, j}$, where $x, y, z$ are constants, and the initial conditions are defined along the first…

We revisit an infinitely nested radical by Ramanujan. Utilizing the full strength of his method, we shall arrive at some new infinitely nested radicals.

We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of…

The study of the flag variety $\mathrm{Fl}_n$ and its subvarieties, including Schubert and Richardson varieties, plays a fundamental role in algebraic geometry and algebraic combinatorics. In this paper, we introduce and develop the theory…

We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a…

A classical theorem of Ahlswede and Katona determines the maximum density of the $2$-edge star in a graph with a given edge density. Motivated by its application in hypergraph Tur\'{a}n problems, we establish a refinement of their result…

Consider two $n$-vertex graphs $X$ and $Y$, where we interpret $X$ as a social network with edges representing friendships and $Y$ as a movement graph with edges representing adjacent positions. The friends-and-strangers graph…

Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain…

In this paper, we review the construction and large $N$ study of the continuous two-dimensional Yang--Mills theory with gauge group $\mathrm{U}(N)$ through probability, combinatorics and representation theory. In the first part, we define…

The localization game is a variant of the game of Cops and Robber in which the robber is invisible and moves between adjacent vertices, but the cops can probe any $k$ vertices of the graph to obtain the distance between probed vertices and…

We develop an analytic approach that draws on tools from Fourier analysis and ergodic theory to study Ramsey-type problems involving sums and products in the integers. Suppose $Q$ denotes a polynomial with integer coefficients. We establish…