Related papers: Strongly sublinear separators and bounded asymptot…

A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that…

We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges,…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic…

For $S \subset \mathbb{R}^n$ and $d > 0$, denote by $G(S, d)$ the graph with vertex set $S$ with any two vertices being adjacent if and only if they are at a Euclidean distance $d$ apart. Deem such a graph to be ``non-trivial" if $d$ is…

A graph $G$ with $d+1$ distinct eigenvalues is called strongly distance-regular if $G$ itself is distance-regular, and its distance-$d$ graph $G_d$ is strongly-regular. In this note we provide a spectral characterization of those…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…

Asymptotic dimension and Assouad-Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott [J. Eur. Math. Society] showed that any proper minor-closed class has…

We consider instances of long-range percolation on $\mathbb Z^d$ and $\mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the…

We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any measurable set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the…

Let $B(2d-1, d)$ be the subgraph of the hypercube $\mathcal{Q}_{2d-1}$ induced by its two largest layers. Duffus, Frankl and R\"odl proposed the problem of finding the asymptotics for the logarithm of the number of maximal independent sets…

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that…

The Planar Separator Theorem, which states that any planar graph $\mathcal{G}$ has a separator consisting of $O(\sqrt{n})$ nodes whose removal partitions $\mathcal{G}$ into components of size at most $\tfrac{2n}{3}$, is a widely used tool…

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most $d$ if it admits an elimination…

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…

Given positive integers $k\leq d$ and a finite field $\mathbb{F}$, a set $S\subset\mathbb{F}^{d}$ is $(k,c)$-subspace evasive if every $k$-dimensional affine subspace contains at most $c$ elements of $S$. By a simple averaging argument, the…

Let G be a finite group acting orthogonally on a pair (S^d,\Gamma) where \Gamma is a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y.…

We prove that $(1+o(1))\sqrt{e/n}$ is the sharp threshold for the appearance of the square of a Hamilton cycle in $G(n,p)$, confirming the conjecture of Kahn, Narayanan, and Park. We also find the exact asymptotics of the threshold for the…

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…