Related papers: Packing Disks by Flipping and Flowing
To investigate the topological structure of Morse flows on the 2-disk we use the planar graphs as destinguished graph of the flow. We assume, that the flow is transversal to the boundary of the 2-disk. We give a list of all planar graph…
We consider the incompressible two-dimensional Euler equation in the plane in the case when its initial vorticity is the characteristic function of a bounded open set. We show that the travel distance grows linearly for most of fluid…
A graph $G=(V,E)$ is a $k$-critical graph if $G$ is not $(k -1)$-colorable but $G-e$ is $(k-1)$-colorable for every $e\in E(G)$. In this paper, we construct a family of 4-critical planar graphs with $n$ vertices and $\frac{7n-13}{3}$ edges.…
Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a…
Given graphs $F$ and $G$, a perfect $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$ that together cover all the vertices in $G$. The study of the minimum degree threshold forcing a perfect $F$-tiling in a graph $G$…
We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We…
If an appropriate region of Kerr-Newman space-time is removed and suitable identifications are made, the resulting space-time can be interpreted as an infinitely thin disk producing the original electromagnetic and gravitational fields. We…
We develop a new numerical method for thin plates falling in inviscid fluid that allows for leading-edge vortex shedding. The inclusion of leading-edge shedding restores physical dynamics to vortex-sheet models of falling bodies, and for…
We adapt the Ozsv\'ath-Szab\'o full path algorithm to every star-shaped graph and establish a correspondence between negative-twisting tight contact structures on any Seifert fibred space over $S^2$, and its Heegaard Floer homology groups…
Directions of graphs were originally introduced in the study of a cops-and-robbers kind of game, while the study of end spaces has been used to generalize classical graph-theoretical results to infinite graphs, such as Halin's…
The interaction of charged particles, moving in a uniform magnetic field, with a plane-polarized gravitational wave is considered using the Fokker-Planck- Kolmogorov (FPK) approach. By using a stochasticity criterion, we determine the exact…
Hyperbolic inversive distance circle packings on the $2$-sphere correspond to Koebe polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic $3$-space. Koebe polyhedra are triangulated convex hyperbolic polyhedra with hyperideal…
A graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion.…
Finite-sheeted covering mappings onto compact connected groups are studied. It is shown that a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected abelian group G must be a homeomorphism…
Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for…
For a set $\mathcal{D}$ of disks in the plane, its disk graph $G(\mathcal{D})$ is the graph with vertex set $\mathcal{D}$, where two vertices are adjacent if and only if the corresponding disks intersect. Given a set $\mathcal{D}$ of $n$…
The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical…
A classic theorem of Erd\H{o}s and P\'osa (1965) states that every graph has either $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices meeting all its cycles. While the standard proof revolves around finding a large `frame' in…
A leaf matching operation on a graph consists of removing a vertex of degree~$1$ together with its neighbour from the graph. For $k\geq 0$, let $G$ be a $d$-regular cyclically $(d-1+2k)$-edge-connected graph of even order. We prove that for…
Current theory of disk galaxy formation is used to study fundamental-plane (FP) type of relations for disk galaxies. We examine how the changes in model parameters affect these relations and explore the possibility of using such relations…