组合数学

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fr\"oberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic…

Let $[0, 1]^{n} \subseteq \mathbb{R}^{n}$ be endowed with its pointwise order, and let $k$ be a positive integer. A subset $A$ of $[0, 1]^{n}$ is said to be a \emph{$k$-antichain} if $\operatorname{card}(A \cap C) \leq k$ for each chain $C…

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device…

Maranca and Rosenberg (2024) devised a ranking scheme for unlabeled multifurcating rooted trees, in which the trees are bijectively associated with the positive integers. Here, generalizing earlier results for bifurcating trees, we…

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the…

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $\beta(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$. While $\beta(G)$ is known for several…

We study the coupon collector's problem in a generalized setting where each draw reveals a fixed number of coupons and the sampling mechanism is required to be \emph{fair}, meaning that every coupon appears with the same frequency among the…

A graph $G$ is $\tau_k$-maximal if $G$ contains no subgraph admitting $k+1$ edge-disjoint spanning trees, while the addition of any edge in the complement of $G$ yields a subgraph that admits $k+1$ edge-disjoint spanning trees. In this…

The largest matching root $\mu(G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest…

We prove that there exists a finite unit-distance graph in the plane with independence ratio strictly smaller than 1/4, answering a question of Erd\H{o}s. Our proof closely follows the framework of Matolcsi, Ruzsa, Varga, and Zs\'amboki,…

The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal degree matrix and the adjacency matrix of $G$, respectively. The signless Laplacian spectral radius of $G$ is the largest eigenvalue…

Let $\mathcal Y(z;t)$ be the modified Greaves--Jing--Zhu operator on the odd power-sum ring. We first point out that this operator can be obtained from the classical neutral operator by a simple diagonal change of variables. We then study…

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a space. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective…

Erd\H{o}s conjectured that every triangle-free graph on $N$ vertices can be made bipartite by deleting at most $N^2/25$ edges; the bound would be sharp, attained by the balanced blow-up $C_5[N/5]$. Writing $\beta(G)$ for the minimum number…

Graphs, maps on surfaces, and abstract polytopes are related combinatorial structures that tend to be studied by different communities using their own tools and databases. Maniplexes provide a unifying framework that captures all of them. A…

This article investigates a remarkable combinatorial identity involving a distinguished family of matrices whose entries are defined via binomial coefficients. Specifically, we consider a class of \( n \times n \) matrices parameterized by…

A quantum Latin square of order \(n\), denoted by \(\operatorname{QLS}(n)\), is an \(n \times n\) square whose entries are unit column vectors in the \(n\)-dimensional Hilbert space \(\mathcal{H}_n\), such that each row and each column…

Negami's Planar Cover Conjecture asserts that a connected graph has a finite planar cover if and only if it can be embedded on the projective plane. While this statement has already been proven for rotation compatible planar covers, namely…

Enami and Maezawa give a complete characterization of $(s_1, s_2, \ldots, s_k)$-linked planar graphs for any $k$-tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show…

Let $R(z)=\sum_{n=0}^{\infty} r_n z^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z)^m$ is unbounded. The proof combines Parseval's identity with Jensen's…