Related papers: A Quasilinear-Time Algorithm for Tiling the Plane …

In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $\Delta\ge 8$ can be edge-colored using $\Delta$ colors. The direct implementation of the Vizing's proof gives an algorithm that finds the coloring in…

Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an $O(n^{4/3}\log^{5/3}n\log^{O(1)}\log n)$-time…

If a (cusped) surface S admits an ideal triangulation T with no shears, we show an efficient algorithm to give S as a quotient of hypebolic plane by a subgroup of PSL(2, Z). The algorithm runs in time O(n log n), where n is the number of…

We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new…

Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral…

If every vertex in a map has one out of two face-cycle types, then the map is said to be $2$-semiequivelar. A 2-uniform tiling is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. Clearly, a…

We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a…

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of…

We consider the problem of cutting a set of edges on a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this…

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that no convex polygon with more than three sides can be tiled with…

We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices $(B_1, \dots, B_m)$ and $(C_1, \dots, C_m)$, whether there exists an…

We generalize the classical K\"onig's and B\"ottcher's Theorems in complex dynamics to certain quasiregular mappings in the plane. Our approach to these results is unified in the sense that it does not depend on the local injectivity, or…

A (possibly denerate) drawing of a graph $G$ in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a straight-line drawing of a planar graph…

In this paper, we give polynomial-time algorithms that can take a graph G with a given combinatorial embedding on an orientable surface S of genus g and produce a planar drawing of G in R^2, with a bounding face defined by a polygonal…

Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual…

Given a collection $\{Z_1,Z_2,\ldots,Z_m\}$ of $n$-sided polygons in the plane and a query polygon $W$ we give algorithms to find all $Z_\ell$ such that $W=f(Z_\ell)$ with $f$ an unknown similarity transformation in time independent of the…

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to…

A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a…