Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675

Poincar\'e profiles of groups and spaces

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican

Largest reduced neighborhood clique cover number revisited

Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hat\beta}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hat\beta}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hat\beta}_t(G)$ including the connections to the greatest reduced average density of $G$, or $\bigtriangledown_t(G)$, introduce the class of graphs with bounded neighborhood clique cover number, and derive a simple lower and an upper bound for this important graph parameter. We announce two conjectures, one for the value of ${\hat\beta}_t(G)$, and another for a separator theorem (with respect to a certain measure) for an interesting class of graphs, namely the class of incomparability graphs which we suspect to have a polynomial bounded neighborhood clique cover number, when the size of a largest induced star is bounded.Comment: The results in this paper were presented in 48th Southeastern Conference in Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, March 201

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices)