Universal graphs with a forbidden subtree

We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists

A continuum of expanders

A regular equivalence between two graphs $\Gamma,\Gamma'$ is a pair of uniformly proper Lipschitz maps $V\Gamma\to V\Gamma'$ and $V\Gamma'\to V\Gamma$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.Comment: 11 pages, accepted for publication in Fundamenta Mathematica

Schnyder decompositions for regular plane graphs and application to drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with faces of degree $d$) for all $d\geq 3$. A \emph{Schnyder decomposition} is a set of $d$ spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly $d-2$ of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the $d$-angulation is $d$. As in the case of Schnyder woods ($d=3$), there are alternative formulations in terms of orientations ("fractional" orientations when $d\geq 5$) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed $d$-angulation of girth $d$ is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on $d$-regular plane graphs of mincut $d$ rooted at a vertex $v^*$) are decompositions into $d$ spanning trees rooted at $v^*$ such that each edge not incident to $v^*$ is used in opposite directions by two trees. Additionally, for even values of $d$, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph $G$ of mincut 4 with $n$ vertices plus a marked vertex $v$, the vertices of $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ has exactly one bend. Embedding also the marked vertex $v$ is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to $v$. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices

Expanders Are Universal for the Class of All Spanning Trees

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i.e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and \Delta, we denote by T(n,\Delta) the class of all n-vertex trees with maximum degree at most \Delta. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is T(n,\Delta)-universal or, in other words, contains every spanning tree of maximum degree at most \Delta. Our methods also apply to the case when \Delta is some function of n. The result has a few very interesting implications. Most importantly, we obtain that the random graph G(n,p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded degree spanning (i.e., n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a constant. Moreover, a corresponding result holds for the random regular graph of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta, we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs with clique number at most five. Finally, we show robustness of random graphs with respect to being universal for T(n,\Delta) in the context of the Maker-Breaker tree-universality game.Comment: 25 page

Learning loopy graphical models with latent variables: Efficient methods and guarantees

The problem of structure estimation in graphical models with latent variables is considered. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider models where the underlying Markov graph is locally tree-like, and the model is in the regime of correlation decay. For the special case of the Ising model, the number of samples $n$ required for structural consistency of our method scales as $n=\Omega(\theta_{\min}^{-\delta\eta(\eta+1)-2}\log p)$, where p is the number of variables, $\theta_{\min}$ is the minimum edge potential, $\delta$ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and $\eta$ is a parameter which depends on the bounds on node and edge potentials in the Ising model. Necessary conditions for structural consistency under any algorithm are derived and our method nearly matches the lower bound on sample requirements. Further, the proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the dimension growth of groups

Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections between dimension growth, expansion properties of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only prove that the lower bound of the dimension growth of $F$ is exp sqrt(n). New open questions and comments are added. v4: The paper is completely revised. Dimension growth with control is introduced, connections with graph expansion and Ramsey theory are include