组合数学
Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erd\H{o}s-Ko-Rado Theorem to the…
A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine…
We give a counterexample to a conjecture made by Cigler, Jerman and Wojciechowski stating that all posets are conclusive. We also provide combinatorial characterizations for conclusiveness of finite posets and the existence of outer…
Erd\H{o}s, S\'ark\"ozy, and S\'os studied the asymptotics of the maximum size of a subset of $\{1,2,\ldots, N\}$ such that it does not contain $k$ distinct elements whose product is a perfect square. More generally, Verstra\"ete proposed a…
Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the…
In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose…
Shift graphs, introduced by Erd\H{o}s and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a suprising property: for each…
We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We…
The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and…
In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the K\"ahler package such that their Hilbert--Poincar\'e series agrees with the Chow polynomial. Our statement provides…
The transmission of a vertex $v$ in a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. The Wiener index $W(G)$ of a…
Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $\Pi_F$ which can be used to provide lower bounds on the Hamming distance between a…
D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of…
A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. A graph $G$ is perfectly weight divisible if for every positive…
Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. A configuration $C$ is a supply of pebbles at various vertices of a…
The spectral radius of a graph is the largest modulus of an eigenvalue of its adjacency matrix. Let $\mathcal{C}_{n, e}$ be the set of all the connected simple graphs with $n$ vertices and $n - 1 + e$ edges. Here, we solve the spectral…
A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for…
Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this…
We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result…
Given a class of (undirected) graphs $\mathcal{C}$, we say that a Feedback Arc Set (FAS for short) $F$ is a $\mathcal{C}$-FAS if the graph induced by the edges of $F$ (forgetting their orientations) belongs to $\mathcal{C}$. We show that…