Classifying Convex Bodies by their Contact and Intersection Graphs

The strong chromatic index of a graph , denoted by , is defined as the least number of colors in a coloring of edges of , such that each color class is an induced matching (or: if edges and have the same color, then both vertices of are not adjacent to any vertex of ). A graph is a unit distance graph in if vertices of can be uniquely identified with points in , so that is an edge of if and only if the Euclidean distance between the points identified with and is 1. We would like to find the largest possible value of , where is a unit distance graph (in and ) of maximum degree . We show that , where is a unit distance graph in of maximum degree . We also show that the maximum possible size of a strong clique in unit distance graph in is linear in and give a tighter result for unit distance graphs in the plane.

In this paper, we obtain a lower bound on the number of edges in a unit distance graph Γ in an infinitesimal plane layer ℝ2 × [0, e]d, which relates the number of edges e(Γ), the number of vertices ν(Γ), and the independence number α(Γ). Our bound $$ e\left(\varGamma \right)\ge \frac{19\nu \left(\varGamma \right)-50\alpha \left(\varGamma \right)}{3} $$ is a generalization of a previous bound for distance graphs in the plane and a strong improvement of Turan’s bound in the case where $$ \frac{1}{5}\le \frac{\alpha \left(\varGamma \right)}{v\left(\varGamma \right)}\le \frac{2}{7} $$ .

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR 2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a “unit distance graph” inR 2d.

Known realizations of geometric representations of graphs, like contact, intersection, etc., are “continuous”, in the sense that the geometric objects are drawn in Euclidean space with real numbers as coordinates. In this paper, we initiate the study of dicrete versions of contact and intersection graphs and examine their relation to their continuous counterparts. The classes of graphs arising appear to have interesting properties and are thus interesting in their own right. We also study realizability, characterizations as well as intractability questions for the resulting new classes of graphs.1980 Mathematics Subject Classification68111068U05CR CategoriesF.2.2Key Words and PhrasesCoinContactIntersectionInterval graphsDiscretePlanar graphsNP

Products of unit distance graphs

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hormander-R»dstrom lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.

Classes and Recognition of Curve Contact Graphs

Graph databases in many applications - semantic web, transport or biological networks among others - are not only large, but also frequently modified. Evaluating graph queries in this dynamic context is a challenging task, as those queries often combine first-order and navigational features. Motivated by recent results on maintaining dynamic reachability, we study the dynamic evaluation of traditional query languages for graphs in the descriptive complexity framework. Our focus is on maintaining regular path queries, and extensions thereof, by first-order formulas. In particular we are interested in path queries defined by non-regular languages and in extended conjunctive regular path queries (which allow to compare labels of paths based on word relations). Further we study the closely related problems of maintaining distances in graphs and reachability in product graphs. In this preliminary study we obtain upper bounds for those problems in restricted settings, such as undirected and acyclic graphs, or under insertions only, and negative results regarding quantifier-free update formulas. In addition we point out interesting directions for further research.

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon<1$. Let $G_\epsilon$ be the graph whose vertex set is the Euclidean plane, where two vertices are adjacent iff the Euclidean distance between them lies in the closed interval $[1-\epsilon,1+\epsilon]$. What is the chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ is precisely the classical ``chromatic number of the plane'' problem. In a 2018 preprint, de Grey shows that $\chi(G_0)\geq 5$; the proof relies heavily on machine computation. In 2016, Grytczuk et al. proved a weaker result with a human-comprehensible but nonconstructive proof: whenever $0<\epsilon<1$, we have that $\chi(G_\epsilon)\geq 5$. (This lower bound of $5$ was later improved by Currie and Eggleton to $6$.) The De Bruijn - Erd\H{o}s theorem (which relies on the axiom of choice) then guarantees the existence, for each $\epsilon$, of a finite subgraph $H_\epsilon$ of $G_\epsilon$ such that $\chi(H_\epsilon)\geq 5$. In this paper, we explicitly construct such finite graphs $H_\epsilon$. We find that the number of vertices needed to create such a graph is no more than $2\pi(15+14\epsilon^{-1})^2$. Our proof can be done by hand without the aid of a computer.

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Toth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

We propose a new robust algorithm for Boolean operations on solid models. The algorithm produces a consistent intersection graph between two input solids whose geometrical data are represented in floating point numbers. In order to prevent numerical calculation errors and inaccuracy of input data from causing inconsistency of the output, we put higher priority on symbolical connectivity of the edge‐face intersection points than their numerical nearness. Each edge‐face intersection point is symbolically represented using face names, which generate connectivity relations between the intersection points and the intersection line segments. The symbols with the same connectivity are made into clusters. The intersection line segments connected together at their end clusters form the intersection graph of two solids. Inconsistency of the connectivity of the clusters is detected and the intersection graph is corrected automatically. We describe the algorithm in detail for polyhedral solids, discuss extension to curves solids, and show its effectiveness by some examples of Boolean operations for two solids whose faces intersect at a very small angle.

OF THE DISSERTATION Some Problems On Discrete Geometry and Combinatorics by Lei Wang Dissertation Directors: William Steiger and Mario Szegedy Let K be a convex body in the plane. It is known that K can never be partitioned into seven regions of equal area by three non-concurrent lines. We will be concerned with a partition of K by three non-concurrent lines such that the ratio of the area of smallest region to the area of biggest region is maximum. We call this an optimal balanced partition at K. We show that the best possible ratio is achieved when K is a triangle and we characterize the optimal balanced partition in this case. We conjecture that the condition holds for optimal balanced partitions of all convex bodies but can only prove a weaker result. In the second part of the thesis, we switch to the zigzag problem. We are given a set of n points in R2 and seek the minimum number of line segments required for a polygonal chain (or a simple polygonal chain) to traverse all the points. We show an n/2 + O(n/ log n) upper bound if self-intersection is allowed and an n − ⌊n−2 8 ⌋ upper bound if self-intersection is not allowed. The third part of this thesis is about finding the optimally balanced forward degree sequence of a graph. The final part studies the optimal solutions for some variants of the Towers of Hanoi problem.

In [2], we have shown that the extended diameter family of a convex body C in n-space has the property of outward simplicity if and only if there is no pair of parallel line segments in the boundary of C respectively in an opposite pair of parallel contact planes of C. In [3] Professor Sobczyk and I show that every outwardly simple family of lines in the plane is the extended diameter family of a constant breadth convex body and incidentally give a simple construction for all constant breadth planar convex bodies. The purpose of this note is to show that not every continuous outwardly simple line family in Euclidean 3-space is the extended diameter family of a convex body. A diameter of a convex body in En (n > 2) is a longest chord in its direction. In [2] it is proved that the diameters of a convex body cover the body.

The Unit-Distance Graph problem in Euclidean plane asks for the minimum number of colors, so that each point on the Euclidean plane can be assigned a single color with the condition that the points at unit distance apart are assigned different colors. It is well known that this number is between 4 and 7, but the exact value is not known. Here this problem is generalized to Minkowski metric spaces and once again the answer is shown to be between 4 and 7. In extreme special cases where the unit circle is a parallelogram or a hexagon the answer is shown to be exactly 4.

Let F=F(n) denote the largest number such that any set of n points in the plane with minimum distance 1 has at least F elements, no two of which are at unit distance. Improving a result by Pollack, we show that F(n) ź 9/35n. We also give an O(n log n) time algorithm for selecting at least 9/35n elements in a set of n points with minimum distance 1 so that no two selected points are at distance 1.