Related papers: Large expanders in high genus unicellular maps
We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope…
We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary 'critical' region containing a…
We describe a new family of $k$-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree $2$, even when the edge density (number of edges…
We study the combinatorial geometry of a random closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. We prove that primitive components of a random multicurve represent linearly independent…
Most statistical models for networks focus on pairwise interactions between nodes. However, many real-world networks involve higher-order interactions among multiple nodes, such as co-authors collaborating on a paper. Hypergraphs provide a…
Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to…
In the hyperbolic plane there are infinite regular lattices. From a fix vertex of a lattice tree graphs can be constructed recursively to the next layers with edges of the lattice. In this article we examine the properties of the growing of…
On a closed symplectic surface Sigma of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Sigma) to the cohomology group H^1(Sigma;R) that extends the flux…
We consider random hyperbolic graphs in hyperbolic spaces of any dimension $d+1\geq 2$. We present a rescaling of model parameters that casts the random hyperbolic graph model of any dimension to a unified mathematical framework, leaving…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…
Let $\Delta \ge 3$ be fixed, $n \ge n_\Delta$ be a large integer. It is a classical result that $\Delta$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c…
A family of models of growing hypergraphs with preferential rules of new linking is introduced and studied. The model hypergraphs evolve via the hyperedge-based growth as well as the node-based one, thus generalizing the…
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is known to be exponentially distributed in Euclidean and real hyperbolic space. We show that the same phenomenon holds on every…
We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic…
It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various…
In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends,…
Locally finite self-similar graphs with bounded geometry and without bounded geometry as well as non-locally finite self-similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume…