On the Computational Complexity of Degenerate Unit Distance
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The Unit Distance Graph Problem and Equivalence Relations
Math/CCS 103 Professor: Padraic Bartlett Lecture 11: The Unit Distance Graph Week 8 UCSB 2014 (Source: \The Mathematical Coloring Book," by Alexander Soifer.) 1 The Unit Distance Graph Problem 2 Definition. Consider the following method for turning R into a graph: 2 • Vertices: all points in R . • Edges: connect any two points (a; b) and (c; d) iff the distance between them is exactly 1. This graph is called the unit distance graph. Visualizing this is kinda tricky | it's got an absolutely insane number of vertices and edges. However, we can ask a question about it: Question. How many colors do we need in order to create a proper coloring of the unit distance graph? So: the answer isn't immediately obvious (right?) Instead, what we're going to try to do is just bound the possible answers, to get an idea of what the answers might be. How can we even bound such a thing? Well: to get a lower bound, it suffices to consider finite graphs G that we can draw in the plane using only straight edges of length 1. Because 2 our graph on R must contain any such graph \inside" of itself, examining these graphs will give us some easy lower bounds! So, by examining a equilateral triangle T , which has χ(T ) = 3, we can see that 2 χ(R ) ≥ 3: This is because it takes three colors to color an equilateral triangle's vertices in such a way that no edge has two endpoints of the same color. Similarly, by examining the following pentagonal construction (called a Moser spindle,) 1 we can actually do one better and say that 2 χ(R ) ≥ 4: Verify for yourself that you can't color this graph with three colors! 2 Conversely: to exhibit an upper bound on χ(R ) of k, it suffices to create a way of \painting" the plane with k-colors in such a way that no two points distance 1 apart get the same color. -
SAT Approach for Decomposition Methods
SAT Approach for Decomposition Methods DISSERTATION submitted in partial fulfillment of the requirements for the degree of Doktorin der Technischen Wissenschaften by M.Sc. Neha Lodha Registration Number 01428755 to the Faculty of Informatics at the TU Wien Advisor: Prof. Stefan Szeider Second advisor: Prof. Armin Biere The dissertation has been reviewed by: Daniel Le Berre Marijn J. H. Heule Vienna, 31st October, 2018 Neha Lodha Technische Universität Wien A-1040 Wien Karlsplatz 13 Tel. +43-1-58801-0 www.tuwien.ac.at Erklärung zur Verfassung der Arbeit M.Sc. Neha Lodha Favoritenstrasse 9, 1040 Wien Hiermit erkläre ich, dass ich diese Arbeit selbständig verfasst habe, dass ich die verwen- deten Quellen und Hilfsmittel vollständig angegeben habe und dass ich die Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken oder dem Internet im Wortlaut oder dem Sinn nach entnommen sind, auf jeden Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht habe. Wien, 31. Oktober 2018 Neha Lodha iii Acknowledgements First and foremost, I would like to thank my advisor Prof. Stefan Szeider for guiding me through my journey as a Ph.D. student with his patience, constant motivation, and immense knowledge. His guidance helped me during the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor. Next, I would like to thank my co-advisor Prof. Armin Biere, my reviewers, Prof. Daniel Le Berre and Dr. Marijn Heule, and my Ph.D. committee, Prof. Reinhard Pichler and Prof. Florian Zuleger, for their insightful comments, encouragement, and patience. -
Eindhoven University of Technology BACHELOR on the K-Independent
Eindhoven University of Technology BACHELOR On the k-Independent Set Problem Koerts, Hidde O. Award date: 2021 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain On the k-Independent Set Problem Hidde Koerts Supervised by Aida Abiad February 1, 2021 Hidde Koerts Supervised by Aida Abiad Abstract In this thesis, we study several open problems related to the k-independence num- ber, which is defined as the maximum size of a set of vertices at pairwise dis- tance greater than k (or alternatively, as the independence number of the k-th graph power). Firstly, we extend the definitions of vertex covers and cliques to allow for natural extensions of the equivalencies between independent sets, ver- tex covers, and cliques. -
Geometric Representations of Graphs
1 Geometric Representations of Graphs Laszl¶ o¶ Lovasz¶ Institute of Mathematics EÄotvÄosLor¶andUniversity, Budapest e-mail: [email protected] December 11, 2009 2 Contents 0 Introduction 9 1 Planar graphs and polytopes 11 1.1 Planar graphs .................................... 11 1.2 Planar separation .................................. 15 1.3 Straight line representation and 3-polytopes ................... 15 1.4 Crossing number .................................. 17 2 Graphs from point sets 19 2.1 Unit distance graphs ................................ 19 2.1.1 The number of edges ............................ 19 2.1.2 Chromatic number and independence number . 20 2.1.3 Unit distance representation ........................ 21 2.2 Bisector graphs ................................... 24 2.2.1 The number of edges ............................ 24 2.2.2 k-sets and j-edges ............................. 27 2.3 Rectilinear crossing number ............................ 28 2.4 Orthogonality graphs ................................ 31 3 Harmonic functions on graphs 33 3.1 De¯nition and uniqueness ............................. 33 3.2 Constructing harmonic functions ......................... 35 3.2.1 Linear algebra ............................... 35 3.2.2 Random walks ............................... 36 3.2.3 Electrical networks ............................. 36 3.2.4 Rubber bands ................................ 37 3.2.5 Connections ................................. 37 4 Rubber bands 41 4.1 Rubber band representation ............................ 41 3 4 CONTENTS -
A Proof of Tomescu's Graph Coloring Conjecture
A proof of Tomescu’s graph coloring conjecture Jacob Fox,∗ Xiaoyu He,† Freddie Manners‡ October 23, 2018 Abstract In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic number n−k k ≥ 4 has at most k!(k − 1) proper k-colorings. Recently, Knox and Mohar proved Tomescu’s conjecture for k = 4 and k = 5. In this paper, we complete the proof of Tomescu’s conjecture for all k ≥ 4, and show that equality occurs if and only if G is a k-clique with trees attached to each vertex. 1 Introduction Let k be a positive integer and G = (V, E) be a graph1.A proper k-coloring, or simply a k-coloring, of G is a function c : V → [k] (here [k] = {1, . , k}) such that c(u) 6= c(v) whenever uv ∈ E. The chromatic number χ(G) is the minimum k for which there exists a k-coloring of G. Let PG(k) denote the number of k-colorings of G. This function is a polynomial in k and is thus called the chromatic polynomial of G. In 1912, Birkhoff [5] introduced the chromatic polynomial for planar graphs in an attempt to solve the Four Color Problem using tools from analysis. The chromatic polynomial was later defined and studied for general graphs by Whitney [41]. Despite a great deal of attention over the past century, our understanding of the chromatic polynomial is still quite poor. In particular, proving general bounds on the chromatic polynomial remains a major challenge. The first result of this type, due to Birkhoff [6], states that n−3 PG(k) ≥ k(k − 1)(k − 2)(k − 3) for any planar graph G on n vertices and any real number k ≥ 5. -
Toward a Unit Distance Embedding for the Heawood Graph
Toward a Unit Distance Embedding for the Heawood graph Mitchell A. Harris1 Harvard Medical School Department of Radiology Boston, MA 02114, USA [email protected] Abstract. The unit distance embeddability of a graph, like planarity, involves a mix of constraints that are combinatorial and geometric. We construct a unit distance embedding for H − e in the hope that it will lead to an embedding for H. We then investigate analytical methods for a general decision procedure for testing unit distance embeddability. 1 Introduction Unit distance embedding of a graph is an assignment of coordinates in the plane to the vertices of a graph such that if there is an edge between two vertices, then their respective coordinates are exactly distance 1 apart. To bar trivial embed- dings, such as for bipartite graphs having all nodes in one part located at (0, 0), for the other part at (0, 1), it is also required that the embedded points be dis- tinct. There is no restriction on edge crossing. A graph is said to be unit distance embeddable if there exists such an embedding (with the obvious abbreviations arXiv:0711.1157v1 [math.CO] 7 Nov 2007 employed, such as UD embeddable, a UDG, etc). For some well-known examples, K4 is not UDG but K4 e is. The Moser spindle is UDG, and is also 4 colorable [5], giving the largest− currently known lower bound to the Erd¨os colorability of the plane [3]. The graph K2,3 is not be- cause from two given points in the plane there are exactly two points of distance 1, but the graph wants three; removing any edge allows a UD embedding. -
SZAKDOLGOZAT Probabilistic Formulation of the Hadwiger
SZAKDOLGOZAT Probabilistic formulation of the Hadwiger–Nelson problem Péter Ágoston MSc in Mathematics Supervisor: Dömötör Pálvölgyi assistant professor ELTE TTK Institute of Mathematics Department of Computer Science ELTE 2019 Contents Introduction 3 1 Unit distance graphs 5 1.1 Introduction . .5 1.2 Upper bound on the number of edges . .6 2 (1; d)-graphs 22 3 The chromatic number of the plane 27 3.1 Introduction . 27 3.2 Upper bound . 27 3.3 Lower bound . 28 4 The probabilistic formulation of the Hadwiger–Nelson problem 29 4.1 Introduction . 29 4.2 Upper bounds . 30 4.3 Lower bounds . 32 5 Related problems 39 5.1 Spheres . 39 5.2 Other dimensions . 41 Bibliography 43 2 Introduction In 1950, Edward Nelson posed the question of determining the chromatic number of the graph on the plane formed by its points as vertices and having edges between the pairs of points with distance 1, which is now simply known as the chromatic number of the plane. It soon became obvious that at least 4 colours are needed, which can be easily seen thanks to a graph with 7 vertices, the so called Moser spindle, found by William and Leo Moser. There is also a relatively simple colouring by John R. Isbell, which shows that the chromatic number is at most 7. And despite the simplicity of the bounds, these remained the strongest ones until 2018, when biologist Aubrey de Grey found a subgraph which was not colourable with 4 colours. This meant that the chromatic number of the plane is at least 5. -
Constructing 5-Chromatic Unit Distance Graphs Embedded in the Euclidean Plane and Two-Dimensional Spheres
Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres V.A. Voronov,∗ A.M. Neopryatnaya,† E.A. Dergachev‡ [email protected] June 24, 2021 Abstract This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson–Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not contain the Moser spindle as a subgraph. The construction of 5-chromatic graphs embedded in a sphere at two values of the radius is given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed. 1 Introduction This paper is devoted to one of the questions related to the Nelson–Hadwiger problem on the chromatic number of the plane [15, 36], namely the development of algorithms that allow constructing new examples of 5-chromatic unit distance graphs embedded in a plane or a sphere, based on a given 4-chromatic unit distance subgraph. The chromatic number of a subset of the Euclidean space X ⊆ Rn is the minimum number of colors needed to color X, so that the points u, v at a unit distance apart are arXiv:2106.11824v2 [math.CO] 23 Jun 2021 colored differently, i.e. -
Chromatic Number of Euclidean Plane
Chromatic number of Euclidean plane Kai-Rui Wang ∗ Abstract: If the chromaticnumber of Euclidean planeis largerthan four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads to confirm the chromatic number of the plane about its exact value. Mathematics Subject Classification (2010): 05C10, 05C55, 05D10. Keywords: Euclidean plane; Finite unit distance graphs; The chromatic number of the plane. 1 Introduction One of the most interesting unsolved problems in geometric graph theory or Euclidean Ramsey theory is as below: How many colors are needed to color the plane so that no two points at unit dis- tance are the same color? which is also called the Hadwiger-Nelson problem, named after Hugo Hadwiger and Edward Nelson.[1,2,3] As for the age of the emergence of it, according to Jensen and Toft’s investigation (1995),[4] the problem was first formulated by E. Nelson in 1950, and first published by Gardner in 1960.[3] Hadwiger in 1945 published a related result,[1] showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).[2] A. Soifer (2003,2008,2011) discusses the problem and its origin extensively.[5−8] Although the answer of this problem is unknown,it has been narrowed down to one of the numbers 4, 5, 6 or 7. The lower bound of four follows from a seven-vertex unit arXiv:1507.00025v1 [math.GM] 1 Jul 2015 distance graph with chromatic number four discovered by brothers William and Leo Moser in 1961,[9] named the Moser spindle. -
A Note on the Unit Distance Problem for Planar Configurations with Q
A note on the unit distance problem for planar configurations with Q-independent direction set Mark Herman and Jonathan Pakianathan July 26, 2021 Abstract Let T (n) denote the maximum number of unit distances that a set 2 of n points in the Euclidean plane R can determine with the additional condition that the distinct unit length directions determined by the config- uration must be Q-independent. This is related to the Erd¨os unit distance problem but with a simplifying additional assumption on the direction set which holds “generically”. We show that T (n + 1) − T (n) is the Hamming weight of n, i.e., the number of nonzero binary coefficients in the binary expansion of n, and find a formula for T (n) explicitly. In particular T (n) is Θ(nlog(n)). Furthermore we describe a process to construct a set of n points in the plane with Q-independent unit length direction set which achieves exactly T (n) unit distances. In the process of doing this, we show T (n) is also the same as the maximum number of edges a subset of vertices of size ∞ n determines in either the countably infinite lattice Z or the infinite hypercube graph {0, 1}∞. The problem of determining T (n) can be viewed as either a type of packing or isoperimetric problem. Keywords: Unit distance problem, discrete combinatorics, isoperimetric problems. 2010 Mathematics Subject Classification. Primary: 05D99, 52C10; Sec- ondary: 05C35, 52C35. arXiv:1406.6029v2 [math.MG] 25 Jun 2014 1 Introduction Erd¨os posed the following question [Erd46]: What is the maximum possible number u(n) of unit distances determined by an n-point set in the Euclidean plane? Erd¨os conjectured that u(n)= O(n1+ǫ) for any ǫ> 0 but the best that 4 is currently known is that u(n) = O(n 3 ), see for example [SST84], [CEG90], [AH02] and [Sze97]. -
Distinct Distances in Graph Drawings
Distinct Distances in Graph Drawings Paz Carmi∗ Vida Dujmovi´c∗ Pat Morin∗ David R. Woody Submitted: Apr 24, 2008; Accepted: Aug 13, 2008; Published: Aug 25, 2008 Mathematics Subject Classification: 05C62 (graph representations) Abstract The distance-number of a graph G is the minimum number of distinct edge- lengths over all straight-line drawings of G in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance- number of trees, graphs with no K4−-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that n-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in (log n). To O conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as ∆ increases the existential lower bound on the distance-number of ∆-regular graphs tends to Ω(n0:864138). 1 Introduction This paper initiates the study of the minimum number of distinct edge-lengths in a draw- ing of a given graph1. A degenerate drawing of a graph G is a function that maps the ∗School of Computer Science, Carleton University, Ottawa, Canada ( paz,vida,morin @scs.carleton.ca). Research supported by NSERC. f yDepartment ofgMathematics and Statistics, The University of Melbourne, Melbourne, Australia ([email protected]). Supported by a QEII Research Fellowship. -
Prime Coloring of Unit Distance and Non Crossing Graphs
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 09, SEPTEMBER 2019 ISSN 2277-8616 Prime Coloring Of Unit Distance And Non Crossing Graphs P.Murugarajan ,R.Aruldoss Abstract: In the recent era, the coloring of the graphs is arousing the attention of researchers. Among the mathematical models, coloring technique plays a vital role in a broad range of applications in real life problems and it solves several complexities of computer networks. In this paper, prime coloring and its chromatic number of unit distance and non crossing Graphs are depicted and its results are validated with few theorems Subject Classification Code: 05C12, 05C78, 05C15 Keywords: Prime graph, Vertex Coloring, Prime Coloring, Unit distance graph, Non Crossing Graphs —————————————————--- -- 1. INTRODUCTION: Proof: We use the terminology as Douglas B.West [1].let G= Let the path with n distinct vertices, the bijection (V, E) be neither a Pseudo graph nor a disconnected graph ψ :V( ) { c ,c ,c ,…..c } graph if for each edge and here order of V be | V |= n. In this paper, we discuss 1 2 3 n Prime graphs and Coloring. The Prime labeling emanate e = cicj , gcd{ ψ (ci), ψ (cj)}=1, i≠j, with Entringer et.al 1980 [2,3]. Be the graph G = (V, E) with ψ (ci) =ci vertex set V is said to have a Prime labeling if there exists a ψ (cj)=cj receive distinct colors. Then the path Graph is + + bijection f: V(G) →Z -{0}, Z is the set of positive integers Prime colorable graph. such that for each edge xy E (G ) ,gcd {f(x),f(y)}=1[5].It Corollary 2.3: If path graph is prime then ( P ) 2 .