Related papers: Hyperfiniteness and Borel combinatorics
We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected…
We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph…
We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments.…
Based on a technique of Barvinok and Barvinok and Sober\'on we identify a class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs. Subsequently we give a quasi-polynomial time approximation…
In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching…
We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of…
In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we…
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows…
We characterize the uniqueness condition in the hardcore model for bipartite graphs with degree bounds only on one side, and provide a nearly linear time sampling algorithm that works up to the uniqueness threshold. We show that the…
Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…
We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit…
An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements…
Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order…
This paper consists of two halves. In the first half of the paper, we consider real-valued functions $f$ whose domain is the vertex set of a graph $G$ and that are Lipschitz with respect to the graph distance. By placing a uniform…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length $3$ or $4$. The second…
Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that…
A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed…
Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to…