组合数学

We revisit the Strong Birthday Problem (SBP) introduced by DasGupta'05, which asks for the minimum population n required such that, with a probability of at least 1/2, every individual in the group shares a birthday with at least one other…

Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when…

We investigate various connections between the clustering for the Burrows-Wheeler transform, a lossless algorithm used in data compression, and languages of interval exchange transformations. We show that a primitive word $u$ clusters for a…

This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and…

Zero forcing is a dynamic coloring process on graphs. Initially, each vertex of a graph is assigned a color of either blue or white, and then a process begins by which blue vertices force white vertices to become blue. The zero forcing…

We generalise a sparsity condition for hypergraphs and show a result relating sparseness of hypergraphs to the decomposition of a modified incidence graph into edge-disjoint forests. We also give new sparsity conditions for posets and…

We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/\alpha(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. This result implies that the same bound holds for…

We present a substitution rule for a rhomb tiling with 10-fold rotational symmetry. The tiling is closely related to the Penrose rhomb tilings and can be obtained from the pentagrid construction. We introduce a finite set of marked…

We investigate the Drazin invertibility of adjacency matrices associated with a class of oriented graphs known as oriented Dutch windmill graphs. By analyzing walks of prescribed lengths and exploiting the structure of the minimal…

This article introduces a quantized chip-firing model with close connections to the theory of rational lattice paths and rational parking functions. Given a graph with a sink and positive integers a,b,c with gcd(a,b)=1, a set S of vertices…

A strong edge coloring of a graph $G$ is an edge coloring $\phi\,:\,E(G) \rightarrow \mathbb N$ such that each color class forms an induced matching in $G$. The strong chromatic index of $G$, written $\chi'_s(G)$, is the minimum number of…

For a graph $G$ with $m$ edges, let $\rho(G)$ be its spectral radius, and let $N_F(G)$ denote the number of copies of $F$ in $G$. Nikiforov [Combin. Probab.\,Comput., 2002] proved that for $r\geq 2$, if $\rho(G)>\sqrt{(1-1/r)2m}$, then…

The quadratic embedding constant (QEC) is a numerical invariant associated with quadratic embeddings of graphs into Hilbert spaces, and it is characterized in terms of the distance matrix. For corona graphs $G\odot H$, a general expression…

Optimizing an implicational base of a closure system consists in turning this implicational base into an equivalent one with premises and conclusions as small as possible. This task is known to be hard in general but tractable for a number…

The lower bound for the chromatic number of the Euclidean space of dimension 8 is increased to 25.

A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates…

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this…

For positive integers $\ell$ and $k$, a $(1^\ell, 2^k)$-packing edge-coloring of a graph $G$ is a partition of $E(G)$ into $\ell$ matchings and $k$ induced matchings. A graph is $d$-irregular if it has no adjacent vertices of degree $d$.…

In 1896, Dedekind posed the problem of factoring the group determinant in the non-abelian case to Frobenius, whose solution sparked the birth of finite-group representation theory. Several decades earlier, Cayley introduced the notion of…

We prove that the quantum and classical cluster algebras for all members of the axiomatically defined classes of symmetric quantum and Poisson Cauchon-Goodearl-Letzter extensions possess maximal green sequences in the sense of Keller.…