DOCSLIB.ORG

Fractional Refinements of Integral Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fractional Refinements of Integral Theorems Fractional refinements of integral theorems by Benjamin Richard Moore A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2021 © Benjamin Richard Moore 2021 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Gary MacGillivray Professor, Dept. of Mathematics and Statistics, University of Victoria Supervisor: Luke Postle Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal Member: Sophie Spirkl Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal Member: Bertrand Guenin Professor, Dept. of Combinatorics and Optimization, University of Waterloo Internal-External Member: Ross Willard Professor, Dept. of Pure Mathematics, University of Waterloo ii Author's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions This thesis contains joint work with Logan Grout, Evelyne Smith-Roberge, Richard Brewster, and Douglas B. West. In particular, Chapter 2 is based on work with Logan Grout, Chapter 3 and 4 is based on work with Evelyne Smith Roberge, Chapter 6 is based on work with Richard Brewster, and Chapter 7 is based on work with Douglas B. West. iv Abstract The focus of this thesis is to take theorems which deal with \integral" objects in graph theory and consider fractional refinements of them to gain additional structure. A classic theorem of Hakimi says that for an integer k, a graph has maximum average degree at most 2k if and only if the graph decomposes into k pseudoforests. To find a fractional refinement of this theorem, one simply needs to consider the instances where the maximum average degree is fractional. We prove that for any positive integers k and d, if G has maximum average degree at 2d most 2k + k+d+1 , then G decomposes into k + 1 pseudoforests, where one of pseudoforests has every connected component containing at most d edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of k and d. Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most k if and only if G decomposes into k forests. The Nine Dragon Tree Theorem, proven by Jiang and Yang, provides a fractional refinement of Nash- Williams Theorem. It says, for any positive integers k and d, if a graph G has fractional d arboricity at most k + k+d+1 , then G decomposes into k +1 forests, where one of the forests has maximum degree d. We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let k = 1 d and d 2 f3; 4g. Every graph with fractional arboricity at most 1 + d+2 decomposes into two forests T and F where F has maximum degree d, every component of F contains at most one vertex of degree d, and if d = 4, then every component of F contains at most 8 edges e = xy such that both deg(x) ≥ 3 and deg(y) ≥ 3. 3 In fact, when k = 1 and d = 3, we prove that every graph with fractional arboricity 1+ 5 decomposes into two forests T;F such that F has maximum degree 3, every component of F has at most one vertex of degree 3, further if a component of F has a vertex of degree 3 then it has at most 14 edges, and otherwise a component of F has at most 13 edges. Shifting focus to problems which partition the vertex set, circular colouring provides a way to fractionally refine colouring problems. A classic theorem of Tuza says that ev- ery graph with no cycles of length 1 mod k is k-colourable. Generalizing this to circular colouring, we get the following: Let k and d be relatively prime, with k > 2d, and let s be the element of Zk such that sd ≡ 1 mod k. Let xy be an edge in a graph G. If G − xy is (k; d)-circular-colorable and G is not, then xy lies in at least one cycle in G of length congruent to is mod k for some i in f1; : : : ; dg. If this does not occur with i 2 f1; : : : ; d − 1g, then xy lies in at least two cycles of length 1 mod k and G − xy contains a cycle of length 0 mod k. This theorem is best possible with regards to the number of congruence classes when k = 2d + 1. v A classic theorem of Gr¨otzsch says that triangle free planar graphs are 3-colourable. There are many generalizations of this result, however fitting the theme of fractional re- finements, Jaeger conjectured that every planar graph of girth 4k admits a homomorphism to C2k+1. While we make no progress on this conjecture directly, one way to approach the conjecture is to prove critical graphs have large average degree. On this front, we prove: Every 4-critical graph which does not have a (7; 2)-colouring and is not K4 or W5 17v(G) 5v(G)+2 satisfies e(G) ≥ 10 , and every triangle free 4-critical graph satisfies e(G) ≥ 3 . In the case of the second theorem, a result of Davies shows there exists infinitely many 5v(G)+4 triangle free 4-critical graphs satisfying e(G) = 3 , and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth 5 graphs embeddable on the torus, projective plane, or Klein bottle are 3- colourable. Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a 2- connected bipartite planar graph G with no separating four-cycles and a 3-colouring f, then one can obtain all 3-colourings from f by changing one vertices' colour at a time if and only if G has at most one face of size 6. p We give the natural generalization of this to circular colourings when q < 4. vi Acknowledgements There are many people to thank during my graduate studies. The single best thing to come out of my graduate studies is meeting other graduate students. It is hard to imagine a better graduate student community than what is here in Waterloo. In particular, I would like to thank Kelvin Chan for organizing many parties, baking much bread, planning many camping trips, and watching alot of matroids. I also would like to thank Charupriya Sharma for many very fun times. She was instrumental in making Waterloo fun. Others are also very deserving of individual shout outs, but to do them justice I would have to make the acknowledgements longer than my thesis, and so simply say thanks Alan Wong, Florian Hoersch, Lukas Nabergall, Iain Crump, Sabrina Lato, Ronen Wdowinski, Tim Miller, Martin Pei, Tina Chen, Rose McCarty, Nick Olsen Harris, Matthew Kroeker, Sina Baghal, Amena Assem, Brett Nassaraden, Arnott Kidner, Soffi´a Arnad´ottir,Jason´ Qu, Tao Jiang, James Davies and Kazuhiro Nomoto for being great (and anyone I missed by accident as well!) For this thesis, I would also like to thank two (ex)-graduate students of C and O in particular, Evelyne Smith Roberge and Logan Grout, who contributed greatly to content of this thesis, and without whom it is not clear the thesis would have been completed (at least without many errors). Some profs and postdocs are also deserving of thanks. I would like to thank Carl Feghali for many discussions and opportunities. I would like to thank my undergrad supervisors Rick Brewster and Sean Mcguinness for helping me many times. I would like to thank Naomi Nishimura for the many meetings. I would especially like to thank Joseph Cheriyan for giving me the opportunity to study at Waterloo, introducing me to Logan, having lots of good advice, and also having the good idea of thinking about pseudoforests. I would also like to thank my readers Sophie Spirkl, Bertrand Guenin, Gary MacGilli- vary and Ross Willard for valuable comments on this thesis. I would also like to thank Xuding Zhu, Daqing Yang and Doug West for allowing me to visit Zheijang Normal University, and their many good ideas which helped in the making of this thesis. I would also like to thank Melissa Cambridge for much help throughout my graduate career. Finally I would like to thank NSERC for financial support throughout my graduate and undergraduate career. vii Dedication This thesis is dedicated to all my family members in Kamloops, especially my Mom and Dad. viii Table of Contents List of Figures xi 1 Introduction1 1.1 On the Nine Dragon Tree Theorem......................2 1.2 Colouring sparse graphs............................ 11 1.3 Cycles and paths between colourings..................... 19 1.4 Structure of the Thesis............................. 23 2 The Pseudoforest Strong Nine Dragon Tree Theorem 24 2.1 Introduction................................... 24 2.2 Picking the counterexample.......................... 26 2.3 Augmenting the decomposition........................ 31 2.4 Bounding the maximum average degree.................... 35 3 Strong Dragons are tough to slay 38 3.1 Introduction................................... 38 3.2 Picking the minimal counterexample..................... 38 3.3 Reducible Configurations............................ 44 3.4 The counting argument for non-bad components............... 54 3.5 The counting argument for bad components................. 59 3.6 Strengthening the d = 3; k = 1 case.....................
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • (V, E ) Be a Graph and Let F Be a Function That Assigns to Each Vertex of F:V(G) → {1,2,.....K} Such That for V to a Set of Values from the Set {1,2
    BALANCED DOMINATION NUMBER OF SPECIAL GRAPHS 1S.CHRISTILDA and 2P.NAMASIVAYAM 1Department of Mathematics, Sadakathullah Appa College, Tirunelveli – 627011, Tamil Nadu, INDIA. E-mail: [email protected] 2PG and Research Department of Mathematics, The MDT Hindu College, Tirunelveli – 627010, Tamil Nadu, INDIA. ABSTRACT INTRODUCTION Let G= (V, E) be a graph. A Subset D of V is called a dominating Let G = (V, E ) be a graph and let f set of G if every vertex in V-D is be a function that assigns to each adjacent to atleast one vertex in D. The Domination number γ (G) of G is the vertex of V to a set of values from cardinality of the minimum dominating set of G. Let the set {1,2,.......k} that is, G = (V, E ) be a graph and let f be a function that assigns to each vertex of f:V(G) → {1,2,.....k} such that for V to a set of values from the set {1,2,.......k} that is, f:V(G) → each u,v ϵ V(G), f(u)≠f(v), if u is {1,2,.....k} such that for each u,v V(G), f(u) ≠ f(v), if u is adjacent to v in adjacent to v in G. Then the set D G. Then the dominating set D V (G) V (G) is called a balanced is called a balanced dominating set if In this dominating set if paper, we investigate the balanced domination number for some special graphs. Keywords: balanced, domination, The special graph balanced domination number Mathematics subject classification: 05C69 is the minimum cardinality of the balanced weak balanced dominating set dominating set.
    [Show full text]
  • On the Chromatic Numbers of Planes
    Preliminary Draft revised 3 March, 2010 On the Chromatic Numbers of Planes G. Eric Moorhouse† Dept. of Mathematics, University of Wyoming, Laramie WY 82071, U.S.A. [email protected] Abstract. Define two points of the Euclidean plane R2 to be adjacent if they are at distance 1 from each other. It is known that the chromatic number of the resulting graph satisfies 4 6 χ(R2) 6 7. We obtain some partial results concerning χ(K2) for more general choices of field K. In particular χ(R2)= χ(K2) for some number field K, i.e. Q K R with ⊆ ⊂ [K : Q] < ; moreover points of K2 are constructible by straightedge ∞ 2 and compass from points of K0 where K0 Q is a finite extension of 2 ⊇ 2 odd degree and χ(K0 ) = 2. In the course of studying χ(K ) where K is the real field R or a number field, we are naturally forced to consider the case where K is a finite field. Finally, we pose the problem of deciding whether or not χ(C2) is finite; and similarly χ(K2) for a subfield K C ⊆ containing i. 1. A Field-Theoretic Approach to the Problem Let K be any field (or more generally, any commutative ring with unity). We regard K2 as a graph in which two vertices (a, b), (c,d) K2 are adjacent iff ∈ (1.1) (a c)2 + (b d)2 = 1. − − A natural (and in general very difficult) problem is to determine the chromatic number χ(K2) of this graph. We recall first the relevant definitions: A proper colouring of K2 is a map ψ : K2 (for some set ) such that ψ(a, b) = ψ(c,d) for all (a, b), (c,d) K2 → C C 6 ∈ satisfying (1.1).
    [Show full text]
  • Introduction to Ramsey Theory
    Introduction to Ramsey Theory Lecture notes for undergraduate course Introduction to Ramsey Theory Lecture notes for undergraduate course Veselin Jungic Simon Fraser University Production Editor: Veselin Jungic Edition: Second Edition ©2014-–2021 Veselin Jungic This work is licensed under the Creative Commons Attribution-NonCommercialShareAlike 4.0 International License. Youcan view a copy of the license at http://creativecommons.org/ licenses/by-nc-sa/4.0/. To my sons, my best teachers. - Veselin Jungic Acknowledgements Drawings of Alfred Hale, Robert Jewett, Richard Rado, Frank Ramsey, Issai Schur, and Bartel van der Waerden were created by Ms. Kyra Pukanich under my directions. The drawing of Frank Ramsey displayed at the end of Section 1.3 was created by Mr. Simon Roy under my directions. I am grateful to Professor Tom Brown, Mr. Rashid Barket, Ms. Jana Caine, Mr. Kevin Halasz, Ms. Arpit Kaur, Ms. Ha Thu Nguyen, Mr. Andrew Poelstra, Mr. Michael Stephen Paulgaard, and Ms. Ompreet Kaur Sarang for their comments and suggestions which have been very helpful in improving the manuscript. I would like to acknowledge the following colleagues for their help in creating the PreTeXt version of the text. • Jana Caine, Simon Fraser University • David Farmer, American Institute of Mathematics • Sean Fitzpatrick , University of Lethbridge • Damir Jungic, Burnaby, BC I am particularly thankful to Dr. Sean Fitzpatrick for inspiring and encouraging me to use PreTeXt. Veselin Jungic vii Preface The purpose of these lecture notes is to serve as a gentle introduction to Ramsey theory for those undergraduate students interested in becoming familiar with this dynamic segment of contemporary mathematics that combines, among others, ideas from number theory and combinatorics.
    [Show full text]
  • Cycle Related Graphs on Square Difference Labeling
    Turkish Journal of Computer and Mathematics Education Vol.12 No. 10 (2021), 1482-1486 Research Article Cycle Related Graphs on Square Difference Labeling J. Rashmi Kumar 1 and K. Manimekalai2 1,2 Department of Mathematics, Bharath Institute of Higher Education and Research, Chennai, India. 1Corresponding author: [email protected] Article History Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published online: 28 April 2021 Abstract: In this study, we prove that the graphs cycle 퐶푛 with parallel chords, 2 − 푡푢푝푙푒 of 푍 − 푃푛, Durer-graph, Moser- spindle, Herchel graph are Square difference graph (SDG). Keywords: Square difference Labeling (푆퐷퐿), Z-Pn, Durer-graph, Moser-spindle, Herchel graph, 2-tuple graph. 1.Introduction All graphs in this paper are simple, undirected and finite. We refer J. A. Gallian for detailed study [1] and follow [2] for all terminology and notation. The Square difference labeling is introduced by Shiama [6]. A function of a graph G admits one to one and onto function 푓: 푣(퐺) → {0,1,2, … … 푝 − 1} such that the 1-1 function 푓∗: 퐸(퐺) → 푁 given by 푓∗(푢푣) = | [푓(푢)2 – 푓(푣)2|, for all 푢푣 휖 퐸(퐺), distinct are said to be Square Difference graph[푆퐷퐺] [1,4]. The concept of 2-tuple was introduced by P.L. Vihol [7]. P. Jagadeeswari investigated some various graphs for SDL. [4,5]. In this work, we prove Cycle related graphs on Square difference labeling. We Commenced some preliminaries which are helpful for our work. Definition 1.1[5]: The graph 푍 − 푃푛 is acquired from the two paths 푃푛′ and 푃푛′′.
    [Show full text]