组合数学

Each finite configuration of points in the plane determines a corresponding lattice of noncrossing partitions. When these points form the vertex set of a convex polygon, the associated lattice is the classical noncrossing partition lattice…

We study independent sets in strong powers of circulant graphs using a transfer matrix formulation. The compatibility constraints separate into intra-layer and inter-layer components, yielding a transfer operator that is equivariant under…

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and…

We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates…

The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_\alpha(\mathscr{G})$, is defined as $RD_\alpha(\mathscr{G})=\alpha\,RT_r(\mathscr{G})+(1-\alpha)\,RD(\mathscr{G}), \, \alpha\in[0,1],$ where…

We study the behavior of West's stack-sorting map $s$ on permutations whose last entry is also their least. Let $S_{n}':=\{\pi0\mid \pi\in S_n\}$ where $\pi0$ denotes the concatenation of $\pi$ and $0$. For each permutation $\pi\in S_n'$,…

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps…

In 1984, Andrews introduced the family of partition functions \(c\phi_k(n)\), which counts the number of generalized Frobenius partitions of \(n\) with \(k\) colors. In previous work, we proved a conjecture on congruences for \(c\phi_6(n)\)…

Let $\bar{\Gamma}$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $\sigma$ is an automorphism of…

We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive…

Given the $r$-distance graph on the hypercube $\mathbb{F}_2^n$, where two vertices are adjacent if their Hamming distance is exactly $r$, we study the maximum size $T(n,r)$ of a triangle-free set of vertices. For even $r\le n/2$, we prove…

A dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G) \setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G) \setminus L$ has…

This book studies ultrafilters on connectivity systems, that is, on pairs \((X,f)\) where \(X\) is a finite set and \(f:2^{X}\to \mathbb{N}\) is a symmetric submodular function. Ultrafilters, which play a fundamental role in topology and…

In the Metric Dimension problem, one asks for a minimum-size set $R$ of vertices such that for any pair of vertices of the graph, there is a vertex from $R$ whose two distances to the vertices of the pair are distinct. This problem has…

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently…

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer…

A graph is diameter-2-critical if its diameter is 2 but the removal of any edge increases the diameter. A well-studied conjecture, known as the Murty-Simon conjecture, states that any diameter-2-critical graph of order n has at most…

The present paper deals with connected subtraction games in graphs, which are generalization of takeaway games. In a connected subtraction game, two players alternate removing a connected sub-graph from a given connected game-graph,…