Topics in Topological Graph Theory Edited by Lowell W
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. -
Excluding a Ladder
EXCLUDING A LADDER TONY HUYNH, GWENAEL¨ JORET, PIOTR MICEK, MICHALT. SEWERYN, AND PAUL WOLLAN Abstract. A ladder is a 2 × k grid graph. When does a graph class C exclude some ladder as a minor? We show that this is the case if and only if all graphs G in C admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph H of G, there must be a color that appears exactly once in H. The minimum number of colors in a centered coloring of G is the treedepth of G, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length k has a path of length k + 1. We show that every 3-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a 2 × k grid has a 2 × (k + 1) grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. -
Self-Dual Configurations and Regular Graphs
SELF-DUAL CONFIGURATIONS AND REGULAR GRAPHS H. S. M. COXETER 1. Introduction. A configuration (mci ni) is a set of m points and n lines in a plane, with d of the points on each line and c of the lines through each point; thus cm = dn. Those permutations which pre serve incidences form a group, "the group of the configuration." If m — n, and consequently c = d, the group may include not only sym metries which permute the points among themselves but also reci procities which interchange points and lines in accordance with the principle of duality. The configuration is then "self-dual," and its symbol («<*, n<j) is conveniently abbreviated to na. We shall use the same symbol for the analogous concept of a configuration in three dimensions, consisting of n points lying by d's in n planes, d through each point. With any configuration we can associate a diagram called the Menger graph [13, p. 28],x in which the points are represented by dots or "nodes," two of which are joined by an arc or "branch" when ever the corresponding two points are on a line of the configuration. Unfortunately, however, it often happens that two different con figurations have the same Menger graph. The present address is concerned with another kind of diagram, which represents the con figuration uniquely. In this Levi graph [32, p. 5], we represent the points and lines (or planes) of the configuration by dots of two colors, say "red nodes" and "blue nodes," with the rule that two nodes differently colored are joined whenever the corresponding elements of the configuration are incident. -
Minor-Minimal Non-Projective Planar Graphs with an Internal 3-Separation
MINOR-MINIMAL NON-PROJECTIVE PLANAR GRAPHS WITH AN INTERNAL 3-SEPARATION A Thesis Presented to The Academic Faculty by Arash Asadi Shahmirzadi In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Algorithms, Combinatorics, and Optimization School of Mathematics Georgia Institute of Technology December, 2012 MINOR-MINIMAL NON-PROJECTIVE PLANAR GRAPHS WITH AN INTERNAL 3-SEPARATION Approved by: Dr. Robin Thomas, Advisor Dr. William T. Trotter School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Dr. William Cook Dr. Xingxing Yu School of Industrial and Systems School of Mathematics Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Prasad Tetali Date Approved: December, 2012 School of Mathematics Georgia Institute of Technology To my parents, for educating me in what truly matters. iii TABLE OF CONTENTS DEDICATION .................................. iii LIST OF FIGURES .............................. v SUMMARY .................................... vi I INTRODUCTION ............................. 1 1.1 GraphTheoreticPreliminaries . 1 1.2 GraphsonSurfaces ........................... 4 1.3 Embedding and Excluding Subgraphs and Minors . 5 1.4 Non-Planar Extensions of Planar Graphs . 9 1.5 Application of the list of minor minimal non-projective planar graphs 10 1.6 Previous approaches for finding the list of minor minimal non-projective planargraphs .............................. 13 1.7 Minor Minimal Non-Projective Planar Graphs . .. 15 1.8 MainResults .............................. 16 1.9 OutlineoftheProof .......................... 19 II NON-c-PLANAR EXTENSIONS OF A c-DISK SYSTEM .... 23 2.1 Definitions and Preliminaries . 23 2.2 Usefullemmas.............................. 31 III SOME APPLICATIONS OF THE THEORY TO ROOTED GRAPHS ......................................... 48 IV OBSTRUCTIONS FOR c-, ac-, abc-PLANARITY ......... 66 4.1 Obstructions for c-planarity ..................... -
Planar Emulators Conjecture Is Nearly True for Cubic Graphs
Planar Emulators Conjecture Is Nearly True for Cubic Graphs Martin Derkaa, Petr Hlinˇen´yb,1 aDavid R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada bFaculty of Informatics, Masaryk University, Botanick´a68a, 602 00 Brno, Czech Republic Abstract We prove that a cubic nonprojective graph cannot have a finite planar emu- lator, unless it belongs to one of two very special cases (in which the answer is open). This shows that Fellows' planar emulator conjecture, disproved for general graphs by Rieck and Yamashita in 2008, is nearly true on cubic graphs, and might very well be true there definitely. Keywords: planar emulator; projective planar graph; graph minor 1. Introduction A graph G has a finite planar emulator H if H is a planar graph and there is a graph homomorphism ' : V (H) ! V (G) where ' is locally surjective, i.e. for every vertex v 2 V (H), the neighbours of v in H are mapped surjectively onto the neighbours of '(v) in G. We also say that such a G is planar-emulable. If we insist on ' being locally bijective, we get H a planar cover. The concept of planar emulators was proposed in 1985 by M. Fellows [6], and it tightly relates (although of independent origin) to the better known planar cover conjecture of Negami [11]. Fellows also raised the main question: What is the class of graphs with finite planar emulators? Soon later he conjectured that the class of planar-emulable graphs coincides with the class of graphs with finite planar covers (conjectured to be the class of projective graphs by Negami [11]| still open nowadays). -
Graph Diameter, Eigenvalues, and Minimum-Time Consensus∗
Graph diameter, eigenvalues, and minimum-time consensus∗ J. M. Hendrickxy, R. M. Jungersy, A. Olshevskyz, G. Vankeerbergheny July 24, 2018 Abstract We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called “definitive consensus conjecture" which states that for an undirected connected graph G with diameter D there exist D matrices whose nonzero-pattern complies with the edges in G and whose product equals the all-ones matrix. Our first result is a counterexample to the definitive consensus conjecture, which is the first improvement of the diameter lower bound for linear consensus protocols. We then provide some algebraic conditions under which this conjecture holds, which we use to establish that all distance-regular graphs satisfy the definitive consensus conjecture. 1 Introduction Consensus algorithms are a class of iterative update schemes commonly used as building blocks for distributed estimation and control protocols. The progresses made within recent years in analyzing and designing consensus algorithms have led to advances in a number of fields, for example, distributed estimation and inference in sensor networks [6, 20, 21], distributed optimization [15], and distributed machine learning [1]. These are among the many subjects that have benefitted from the use of consensus algorithms. One of the available methods to design an average consensus algorithm is to use constant update weights satisfying some conditions for convergence (as can be found in [3] for instance). However, the associated rate of convergence might be a limiting factor, and this has spanned a literature dedicated to optimizing the speed of consensus algorithms. -
Some Constructions of Small Generalized Polygons
Journal of Combinatorial Theory, Series A 96, 162179 (2001) doi:10.1006Âjcta.2001.3174, available online at http:ÂÂwww.idealibrary.com on Some Constructions of Small Generalized Polygons Burkard Polster Department of Mathematics and Statistics, P.O. Box 28M, Monash University, Victoria 3800, Australia E-mail: Burkard.PolsterÄsci.monash.edu.au and Hendrik Van Maldeghem1 Pure Mathematics and Computer Algebra, University of Ghent, Galglaan 2, 9000 Gent, Belgium E-mail: hvmÄcage.rug.ac.be View metadata, citation and similarCommunicated papers at core.ac.uk by Francis Buekenhout brought to you by CORE provided by Elsevier - Publisher Connector Received December 4, 2000; published online June 18, 2001 We present several new constructions for small generalized polygons using small projective planes together with a conic or a unital, using other small polygons, and using certain graphs such as the Coxeter graph and the Pappus graph. We also give a new construction of the tilde geometry using the Petersen graph. 2001 Academic Press Key Words: generalized hexagon; generalized quadrangle; Coxeter graph; Pappus configuration; Petersen graph. 1. INTRODUCTION A generalized n-gon 1 of order (s, t) is a rank 2 point-line geometry whose incidence graph has diameter n and girth 2n, each vertex corre- sponding to a point has valency t+1 and each vertex corresponding to a line has valency s+1. These objects were introduced by Tits [5], who con- structed the main examples. If n=6 or n=8, then all known examples arise from Chevalley groups as described by Tits (see, e.g., [6, or 7]). Although these examples are strongly group-related, there exist simple geometric con- structions for large classes of them. -
Dynamic Cage Survey
Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages . -
Minors and Dimension
MINORS AND DIMENSION BARTOSZ WALCZAK Abstract. It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it is proved that posets of bounded height whose cover graphs exclude a fixed topological minor have bounded dimension. This generalizes all the aforementioned results and verifies a conjecture of Joret et al. The proof relies on the Robertson-Seymour and Grohe-Marx graph structure theorems. 1. Introduction In this paper, we are concerned with finite partially ordered sets, which we simply call posets. The dimension of a poset P is the minimum number of linear orders that form a realizer of P , that is, their intersection gives rise to P . The notion of dimension was introduced in 1941 by Dushnik and Miller [3] and since then has been one of the most extensively studied parameters in the combinatorics of posets. Much of this research has been focused on understanding when and why dimension is bounded, and this is also the focus of the current paper. The monograph [24] contains a comprehensive introduction to poset dimension theory. To some extent, dimension for posets behaves like chromatic number for graphs. There is a natural construction of a poset with dimension d, the standard example Sd (see Figure1), which plays a similar role to the complete graph Kd in the graph setting. -
Report (221.0Kb)
MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH Tagungsbericht Graph Theory The organizers of this meeting on graph theory were Reinhard Diestel and Paul Seymour Besides the normal formal lectures the meeting included a number of informal sessions Each session was concerned with a particular area of graph theory and anyone interested was welcome to attend During these informal meetings participants presented results and op en problems concerning the topic and the audience was en couraged to interrupt with questions counterexamples pro ofs etc These sessions resulted in the resolution of a number of conjectures as well as stimulating collab oration outside the structure of the conference The following is a summary of the sessions followed by a collection of abstracts of the formal talks Session on Innite Graphs Convenor Reinhard Diestel Cycle space in lo cally nite innite graphs Bruce Richter asked how the fact that the fundamental cycles of a nite graph form a basis of its cycle space can b e adapted appropriately to innite graphs In the discussion it emerged that endfaithful spanning trees would play a signicant role here and various mo dels based on these were discussed Richters ob jective was to prove a uniquenessofembedding theorem for connected lo cally nite graphs with suitable compactication such as one p oint at innity for every class of ends pairwise not separated by a nite cycle This led to further informal collab oration later in the week Transitive graphs and Cayley graphs Recalling Woesss problem of whether every lo cally -
Lucky Labeling of Certain Cubic Graphs
International Journal of Pure and Applied Mathematics Volume 120 No. 8 2018, 167-175 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/ Lucky Labeling of Certain Cubic Graphs 1 2, D. Ahima Emilet and Indra Rajasingh ∗ 1,2 School of Advanced Sciences, VIT Chennai, India [email protected] [email protected] Abstract We call a mapping f : V (G) 1, 2, . , k , a lucky → { } labeling or vertex labeling by sum if for every two adjacent vertices u and v of G, f(v) = f(u). (v,u) E(G) 6 (u,v) E(G) The lucky number of a graph∈ G, denoted by η∈(G), is the P P least positive k such that G has a lucky labeling with 1, 2, . , k as the set of labels. Ali Deghan et al. [2] have { } proved that it is NP-complete to decide whether η(G) = 2 for a given 3-regular graph G. In this paper, we show that η(G) = 2 for certain cubic graphs such as cubic dia- mond k chain, petersen graph, generalized heawood graph, − G(2n, k) cubic graph, pappus graph and dyck graph. − AMS Subject Classification: 05C78 Key Words: Lucky labeling, petersen, generalized hea- wood, pappus, dyck 1 Introduction Graph coloring is one of the most studied subjects in graph theory. Recently, Czerwinski et al. [1] have studied the concept of lucky labeling as a vertex coloring problem. Ahadi et al. [3] have proved that computation of lucky number of planar graphs is NP-hard. -
Crossing Number Graphs
The Mathematica® Journal B E Y O N D S U D O K U Crossing Number Graphs Ed Pegg Jr and Geoffrey Exoo Sudoku is just one of hundreds of great puzzle types. This column pre- sents obscure logic puzzles of various sorts and challenges the readers to solve the puzzles in two ways: by hand and with Mathematica. For the lat- ter, solvers are invited to send their code to [email protected]. The per- son submitting the most elegant solution will receive a prize. ‡ The Brick Factory In 1940, the Hungarian mathematician Paul Turán was sent to a forced labor camp by the Nazis. Though every part of his life was brutally controlled, he still managed to do serious mathematics under the most extreme conditions. While forced to collect wire from former neighborhoods, he would be thinking about mathematics. When he found scraps of paper, he wrote down his theorems and conjectures. Some of these scraps were smuggled to Paul Erdős. One problem he developed is now called the brick factory problem. We worked near Budapest, in a brick factory. There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected to all the storage yards. The bricks were carried on small wheeled trucks to the storage yards. All we had to do was to put the bricks on the trucks at the kilns, push the trucks to the storage yards, and unload them there. We had a reasonable piece rate for the trucks, and the work itself was not difficult; the trouble was at the crossings.