组合数学

Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and…

A graph $G$ is called $k$-edge hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. Motivated by earlier results of Erd\H{o}s, F\"{o}redi, Kostochka and Luo…

Let \(F\) and \(G\) be \(r\)-uniform hypergraphs, and let \(f_{F,G}(n)\) be the largest integer \(m\) such that every \(n\)-vertex \(G\)-free \(r\)-graph contains an induced \(F\)-free subgraph on \(m\) vertices. We prove that, if…

A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $1$'s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun,…

We extend the analysis of nonrepetitive sequences of Entringer et al. [Journal of Combinatorial Theory, 1974] to relaxations of equality testing under nonstandard equivalence relations, in particular parameterized equivalence and…

In a previous work, we defined (type A) c-Birkhoff polytopes and showed that they were unimodularly equivalent to order polytopes of heap posets. In this note we answer the question: What about type B?

A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lov\'asz proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple…

A graph is non-$r$-partite if its chromatic number exceeds $r$. For an edge-color-critical graph $F$ with $\chi(F)=r+1$, let $\mathrm{ex}_{r+1,\rho}(n,F)$ be the maximum adjacency spectral radius among non-$r$-partite $F$-free graphs of…

Let $\mathcal Q=\{Q_a:a\geq1\}$ be a nested family of finite posets such that $Q_a\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of…

We study finite-field analogues of the Peres--Schlag nonempty-interior problem for product sets. Given \(A\subseteq\mathbb F_p\), we ask when a suitable one-dimensional linear image of \(A^n\) is full; equivalently, when there exist…

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size…

Unit-interval parking functions of length $n$ are enumerated by the Fubini numbers $F_n$ and are in explicit bijection with the ordered set partitions of $[n]$. We use this bijection to single out the unit-interval parking functions whose…

The Chromatic Sum problem asks, given a graph $G$ and an integer $k$, whether $G$ admits a colouring $c$ with sum $\sum_{v\in V}c(v) \leq k$. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs.…

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{{M}}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that…

A strong majority edge-coloring of a graph is an edge-coloring in which, for every edge $e$ and every color $i$, at most half of the edges adjacent to $e$ have color $i$. Such a coloring exists only for graphs with no pendant path of length…

The Boolean lattice $BL_n$, $n\geq 3$, is the graph whose vertex set is the collection of all subsets of $[n]=\{1,2,\ldots,n\}$, where two subsets $U$ and $W$ are adjacent if and only if their symmetric difference has precisely one element.…

We prove the sharp diagonal spectral correlation conjecture of Friedgut, Kahn, Kalai and Keller, proposed in their Fourier-analytic approach to Chv\'atal's conjecture. For every pair of increasing Boolean functions…

Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot)$ be the Lov\'asz--Simonovits $r$-clique density function. For any integers $2\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an…

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might…

List-coloring, introduced independently by Vizing and by Erd\H{o}s, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its…