Related papers: Removable singularities for nonlocal minimal graph…

In this note, we derive a uniqueness theorem for minimal graphs of general codimension under certain restrictions closed related to the convexity (not strict convexity) of the area functional with respect to singular values, improving the…

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete,…

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with…

A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable…

A connected, locally finite graph $\Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $\Gamma$. Define the minimal degree of $G$ as…

Local sets, a graph structure invariant under local complementation, have been originally introduced in the context of quantum computing for the study of quantum entanglement within the so-called graph state formalism. A local set in a…

We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily…

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local…

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the…

We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,\gamma}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.

We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even…

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and…

A graph $G$ is said to be ubiquitous, if every graph $\Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every…

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Clearly, a…

We prove that every locally finite quasi-transitive graph that does not contain $K_\infty$ as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their…

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we…

For nonnegative even kernels $K$, we consider the $K$-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated $K$-nonlocal mean curvature equation in an open set…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

In this paper, we compute universal minimal flows of groups of automorphisms of uncountable $\omega$-homogeneous graphs, $K_n$-free graphs, hypergraphs, partially ordered sets, and their extensions with an $\omega$-homogeneous ordering. We…

Given a finite undirected graph $X$, a vertex is $0$-dismantlable if its open neighbourhood is a cone and $X$ is $0$-dismantlable if it is reducible to a single vertex by successive deletions of $0$-dismantlable vertices. By an iterative…