Related papers: Embedding products of graphs into Euclidean spaces
Token graphs, or symmetric powers of graphs, see \cite{alavi2002survey} and \cite{Fabila-Monroy2012}, are defined on the $k$-combinations of the vertex set of some graph $L$, where edges exist between two such combinations, if their…
Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He…
Twin-width is a width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler…
There are two natural Chern-Simons theories associated with the embedding of a three-dimensional surface in Euclidean space; one is constructed using the induced metric connection -- it involves only the intrinsic geometry, the other is…
The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable…
The pinched sphere is the pseudo-surface $\mathbb{S}^{\circ}_0$ obtained by identifying two distinct points of the sphere. We provide a structural characterization of graphs excluding an $\mathbb{S}^{\circ}_0$-embeddable graph as a minor.…
The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated…
There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization,…
Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zumino's N=2 model in four dimensions, provides manifestly…
The fact that the complete graph $K_5$ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph…
Let $\Delta=\Delta_1\times\ldots\times \Delta_d\subseteq\mathbb{R}^n$, where $\mathbb{R}^n=\mathbb{R}^{n_1}\times\cdots\times\mathbb{R}^{n_d}$ with each $\Delta_i\subseteq\mathbb{R}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove…
Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs into Minkowski spacetime using Multidimensional scaling (MDS). First…
We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E) on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that |E_0|…
Graph packing generally deals with unlabeled graphs. In \cite{EHRT11}, the authors have introduced a new variant of the graph packing problem, called the \textit{labeled packing of a graph}. This problem has recently been studied on trees…
We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2 and an m-dimensional subset S of R^n we also introduce global versions of this discrete…
This paper is the last paper in a series of five papers. Building on earlier papers in this series, we prove an analogue of Kuratowski's characterisation of graph planarity for three dimensions. More precisely, a simply connected…
We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of…
Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group.…
We give a new proof of a theorem of Kleiner-Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the…
Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as…