组合数学

The $s$-blowup of a graph ($s\geq2$) is the $2s$-uniform hypergraph obtained by replacing each vertex with a set of size $s$ and preserving the adjacency relation. In this paper, we define $2s$-weighted graphs and use them to give all…

Let $\Pi_{(b^a)}$ be the locus of unordered set partitions of $[ab]$ with $a$ blocks of size $b$. We embed unordered set partitions of $[n]$ into the affine space $\mathbb{C}^{\binom{[n]}{2}}$ with coordinate ring…

We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament.…

We prove that the leading coefficient of the "error" terms of NRS(2) applied to a cubic polynomial $f(z)$ with starting point $(-\frac{a_1}{a_2}, -\frac{a_1}{a_2})$ are positive-coefficient rational functions in the zeros of $f(z)$. We…

We show that all results stated in [E. Camby, F. Plein, Discrete Appl. Math. 217 (2017) 711-717] are either previously known or incorrect. For example, Camby and Plein claimed to provide counterexamples to the 1995 characterization of…

We introduce the package combinatorics for the software CoCoA. This package provides a data structure and the necessary methods for computing several known enumerative combinatorial invariants.

We investigate and generalise Levine sequences like A011784, A061892 and A061894 and develop an algebraic theory for them. We thereby also cover other fast growing sequences like A014644, which we call golombic due to their strong ties with…

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular…

Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called $\varepsilon$-balanced if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erd\H{o}s and Szemer\'{e}di…

An orientation of a given static graph is called transitive if for any three vertices $a,b,c$, the presence of arcs $(a,b)$ and $(b,c)$ forces the presence of the arc $(a,c)$. If only the presence of an arc between $a$ and $c$ is required,…

Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021),…

For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that…

We give a sufficient condition on a positive integer $m$ for every stratum of a given real toric hyperplane arrangement to contain a rational point of denominator $m$. As a consequence, we give a sufficient condition on $m$ for the degree…

In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card that represents multiplex juggling patterns in a natural bijective way. They conjectured a formula for the generating function for the number of multiplex juggling…

In this paper, we prove an analogue of the Erd\H{o}s-Ko-Rado theorem intersecting families of subspaces in projective Hjelmslev geometries over finite chain rings of nilpotency index 2. We give an example of maximal families that are not…

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in…

We prove several results about matroids and matroidal families associated with rigidity in dimension $2$. In particular, we establish new properties of the generic rigidity matroid family $\mathcal{R}$ and Kalai's hyperconnectivity matroid…

In this paper, we prove that the property of being a grape (in any of its variants) is invariant under Alexander duality. The explicitly determined (simple-)homotopy type of a grape can be transferred to its Alexander dual via Combinatorial…

We present correct proof of G. Chung, Y.-J. Tsay result on partial gossip problem.

Let $R$ be a commutative ring with unity 1, and $ G(V,E)$ be a simple, connected, nontrivial graph. Let $d(a,c)$ be the distance between the vertices $a$ and $c $ in $G$. An undirected zero divisor graph of a ring $R$ is denoted by…