组合数学
We study an extension to directed graphs of the parameter called the $b$-chromatic number of a graph in terms of acyclic vertex colorings: the dib-chromatic number. We give general bounds for this parameter. We also show some results about…
There is strong evidence for the belief that `almost all' finite semigroups, whether we consider multiplication operations on a fixed set or their isomorphism classes, are nilpotent of index 3 (3-nilpotent for short). The only known method…
We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all…
Grid polyominoes form a class of thin polyominoes with one or more holes arranged in a grid-like pattern in the plane. In this paper, we prove that the rook polynomial of grid polyominoes coincides with the h-polynomial of their…
The (weak) chromatic number of a hypergraph $H$, denoted by $\chi(H)$, is the smallest number of colors required to color the vertices of $H$ so that no hyperedge of $H$ is monochromatic. For every $2\le k\le d+1$, denote by $\chi_L(k,d)$…
For a set $A$ let ${{\mathbf {SC}}_A}$ be the poset of simplicial complexes whose vertices are in $A$. For a function $f : A \rightarrow B$ there are functors $ f^{! !}, f^{**}, f^{ii}: {{\mathbf {SC}}_A} \rightarrow {{\mathbf {SC}}_B},…
Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern $\delta_k=k(k-1)\cdots21$ in one-line notation and avoid another pattern $\tau$ of length $4$ in all their cycle forms. There are three cases in total to…
The composition $\mathcal{F} \circ \mathcal{G}$ of two combinatorial classes $\mathcal{F}$ and $\mathcal{G}$ is a standard combinatorial construction and translates into the composition $F(G(z))$ of their corresponding counting generating…
For integers $1\le \ell<k$, the $\ell$-degree Tur\'an density $\pi_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Tur\'an density $\pi_1$ and…
Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this…
The precoloring problem of a graph involves assigning colors to some vertices beforehand, and the objective is to determine whether it can be extended to a proper k-coloring of the entire graph. In 1958, Grotzsch proved that every…
The generalized Honeymoon Oberwolfach Problem (HOP) asks whether it is possible to seat $2n$ participants consisting of $n$ newlywed couples at a conference with $s$ tables of size $2$ and $t$ ''round'' tables of sizes $2m_1, 2m_2, \ldots,…
This paper examines vertex colorings of graphs with constraints on the distribution of colors in vertex neighborhoods. We introduce color 2-switches and color degree matrices. The color degree matrix of a $k$-colored graph is an analog of…
One of the most important classes of even $\Delta$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong…
In this paper, we develop a dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices. As a concrete application, we resolve an open problem posed in 2015 by…
Digraph-defined external difference families were recently introduced as a natural generalization of several well-studied combinatorial objects motivated by cryptography (e.g. external difference families (EDFs) and circular external…
Given a labelled tournament on $[n]$, \emph{inverting} a vertex subset $X$ means reversing every edge with both endpoints in $X$. Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time…
A finite set $ S \subset \mathbb{R} $ is called a Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x \le y $ are distinct, and a weak Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x < y $ are distinct. For a finite set $ A…
We study the total cut complexes and their Alexander duals. The homotopy type of these complexes is calculated for de $p$th power of a cycle with at least $2rn$ vertices where $p\leq r$, solving part of a conjecture of Bayer, Denker,…
We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the…