Related papers: One-ended spanning subforests and treeability of g…
We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel…
We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse…
The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…
On a finite graph, there is a natural family of Boltzmann probability measures on cycle-rooted spanning forests, parametrized by weights on cycles. For a certain subclass of those weights, we construct Gibbs measures in infinite volume, as…
In 2001, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+\gamma\right)n$ contains all spanning trees with maximum degree at most $cn/\log n$. We extend…
Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i)…
Closed subgroups of the group of isometries of the regular tree $\treeq$ that fix an end of the tree and are vertex-transitive are shown to correspond, on one hand, to self-replicating groups acting on rooted trees and, on the other hand,…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs…
Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as asymptotic cones. Applying the results we will find in the context of groups, we will prove that a group with "a few" separable asymptotic cones…
Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in…
We bound the volume of thick embeddings of finite graphs into the Heisenberg group, as well as the volume of coarse wirings of finite graphs into groups with polynomial growth. This work follows the work of Kolmogorov-Brazdin, Gromov-Guth…
We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for…
Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in $K_{a,b}, 2 \leq a \leq b$…
We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this…
Researchers have proposed many methods for fair and robust machine learning, but comprehensive empirical evaluation of their subgroup robustness is lacking. In this work, we address this gap in the context of tabular data, where sensitive…
In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose…
When $\mathcal{C}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When $\mathcal{C}$ is indeed a finite simple graph, we study various characterizations of…