组合数学

A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of…

The famous pancake theorem states that for every finite set $X$ in the plane, there exist two orthogonal lines that divide $X$ into four equal parts. We propose an algorithm whose running time is linear in the number of points in $X$ and…

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…

We establish functional Loomis--Whitney type inequalities in the finite Heisenberg group $\mathbb{H}^n(\mathbb{F}_q)$. For $n=1$, we determine the sharp region of exponents $(u_1,u_2)$ for which the Heisenberg Loomis--Whitney inequality \[…

An \emph{$H$-packing} in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor cn \rfloor$-regular graph $G$ admits an $H$-packing that…

Let $\mathcal{A}$ be an affine hyperplane arrangement, $L(\mathcal{A})$ its intersection poset, and $\chi_{\mathcal{A}}(t)$ its characteristic polynomial. This paper aims to propose combinatorial structures for the factorization of…

We construct a new family of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For this family, we explicitly construct their companion. We also…

Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each…

The independent set sequence of trees has been well studied, with much effort devoted to the (still open) question of Alavi, Malde, Schwenk and Erd\H{o}s on whether the independent set sequence of a tree is always unimodal. Much less…

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of…

In Euclidean Ramsey Theory usually we are looking for monochromatic configurations in the Euclidean space, whose points are colored with a fixed number of colors. In the canonical version, the number of colors is arbitrary, and we are…

Let $p$ be an odd prime and let $E\subset \mathbb{F}_p^2$ with $|E|=p^a$, where $0<a\le 1$. For a direction $V$ (a $1$-dimensional subspace of $\mathbb{F}_p^2$), let $\pi^V:\mathbb{F}_p^2\to \mathbb{F}_p^2/V$ denote the quotient map. We…

We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope…

We say that a subset $M$ of $\mathbb R^n$ is exponentially Ramsey if there are $\epsilon>0$ and $n_0$ such that $\chi(\mathbb R^n,M)\ge(1+\epsilon)^n$ for any $n>n_0$, where $\chi(\mathbb R^n,M)$ stands for the minimum number of colors in a…

Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined…

The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the…

Let $H$ be a multiplicatively written monoid with identity $1_H$ and let $\mathcal{P}_{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set…

A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in…

We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the…

Let $f(n)$ be the maximum sum of the sides of non-overlapping squares (or equilateral triangles) packed inside a unit square or (unit equilateral triangle). In this paper, we explore some properties of $f$ and examine how the square and…