组合数学

The Erd\H{o}s--P\'osa theorem asserts that every graph $G$ with no $k$ disjoint cycles contains a set $X$ of $f(k)$ vertices such that $G\setminus X$ has no cycle. Robertson and Seymour showed that this Erd\H{o}s--P\'osa property also holds…

We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a…

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex planar graph that contains no graph $F \in…

An open question asks whether the chromatic symmetric function (CSF) of a graph distinguishes non-isomorphic trees. While it is known that the CSF does not distinguish unicyclic graphs, examples of pairs of unicyclic graphs with the same…

We construct a diamond-free family in the Boolean lattice whose size is asymptotically larger than the union of two middle layers. Denote the diamond poset by $Q_2$ and let $La(n,Q_2)$ be the maximum size of a family in $2^{[n]}$ containing…

We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top…

A famous conjecture of Anton Kotzig states that for every even integer $n\ge 4$, the complete graph $K_n$ of order $n$ can be decomposed into $n - 1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. Despite…

Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is…

Let $G$ be a simple graph of maximum degree $d$, and let $\mu(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of…

This paper introduces Bernoulli-type and Euler-type Mersenne-Lerch interpolation families associated with Apostol-Mersenne polynomial families. Their construction is based on the Mersenne translation polynomials \(P_{n,M}(x;m)\), defined by…

The dimension of a poset $P$ is the minimum positive integer $d$ such that $P$ is an induced subposet of $\mathbb{R}^d$ equipped with the product order. We give a constant-factor polynomial-time approximation algorithm for computing…

A vertex-transitive closure of $\Gamma$ is a vertex-transitive supergraph of $\Gamma$ on the same vertex set. The vertex-transitive number of a graph $\Gamma$, denoted by $d_{vt}(\Gamma)$, is the smallest integer for which there exists a…

A $\{P_2,P_3\}$-factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is…

We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity…

In this note, we resolve the remaining open case of a conjecture by Han and Zhao concerning the codegree threshold for Hamilton $\ell$-cycles in $k$-uniform hypergraphs. Specifically, we prove that for integers $k\ge 3$, $3k/4\le \ell<k$,…

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Tur\'an density $\gamma(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$…

A homogeneous set in a graph is a clique or a stable set. The Erd\H{o}s--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$.…

A $q$-kernel of a digraph $D$ is an independent set $Q\subseteq D$ such that every vertex of $D$ is reachable from $Q$ by a directed path of length at most $q$, which is a natural generalization of kernels and quasikernels. In this paper,…

A proper sub-box of $A=\{0,1,2\}^d$ is a product $S_1\times\dots\times S_d$ with each $\varnothing\neq S_i\subsetneq\{0,1,2\}$. A double cover is a finite multiset of proper sub-boxes covering every point of $A$ exactly twice; write $f(d)$…

A graph is homogeneously traceable if each vertex is an endpoint of a Hamiltonian path. Chartrand, Gould, and Kapoor (1979) proved irregular homogeneously traceable nonhamiltonian graphs exist for every order $n\ge 9$. Hu and Zhan (DAM,…