DOCSLIB.ORG

A Treatise on Pansophy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Treatise on Pansophy Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2020 A treatise on pansophy Isaac Clarence Wass Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Recommended Citation Wass, Isaac Clarence, "A treatise on pansophy" (2020). Graduate Theses and Dissertations. 18008. https://lib.dr.iastate.edu/etd/18008 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A treatise on pansophy by Isaac Clarence Wass A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Steve Butler, Major Professor Bernard Lidick´y Jack Lutz Ryan Martin Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2020 Copyright c Isaac Clarence Wass, 2020. All rights reserved. ii DEDICATION To all my family, especially the new nieces and nephew, whom I love dearly. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . vi ABSTRACT . vii CHAPTER 1. Introduction . .1 1.1 Motivation and Background . .1 1.2 Definitions . .2 1.3 Previous Results . .4 1.3.1 Observations . .4 1.3.2 Pansophy of Paths and Cycles . .4 CHAPTER 2. General Results . .6 2.1 Further Observations . .6 2.2 Convergence Theorems for Pansophy . .8 2.3 Pansophy of Simple Graph Families . .9 2.3.1 Complete Graphs Minus an Edge . .9 2.3.2 Star Graphs Plus an Edge . 10 2.3.3 A Family of Complete Bipartite Graphs . 11 2.3.4 Triangle Book Graphs . 12 2.3.5 Friendship Graphs . 12 2.3.6 Graphs with Boolean Path-Connection Probabilities . 13 2.4 Pansophy and Graph Operations . 16 2.4.1 Adding Isolated Vertices . 16 2.4.2 Appending Leaves . 17 CHAPTER 3. An Alternate Approach to Computing Pansophy . 21 3.1 The Method . 21 3.2 Wheel Graphs . 23 3.3 Fan Graphs . 27 3.4 Complete Bipartite Graphs . 30 3.5 Miscellaneous Results . 32 iv CHAPTER 4. Graphs with at Most One Cycle . 39 4.1 Forests . 39 4.2 Subunicyclic Graphs . 43 CHAPTER 5. Planar Graphs . 49 5.1 Outerplanar Graphs . 50 5.2 Maximally Planar Graphs . 52 5.3 Planar Graphs in General . 56 CHAPTER 6. Density . 58 6.1 Density Review . 58 6.2 Pansophical Density . 60 6.2.1 In the Interval [0; 1)................................ 60 6.2.2 In the Interval [0; 1] ................................ 62 6.2.3 In the Interval [1; lim Ψ(Pn)]........................... 63 n!1 CHAPTER 7. Computing Pansophy using SAGE . 67 7.1 pk-Sequences For Some Well-Known Graphs . 67 7.2 pk-Sequences For Most Platonic Solids . 68 7.3 pk-Sequences For Some Hypercubes . 69 7.4 pk-Sequence For the Heawood Graph . 70 7.5 Correlation Between Pansophy and Edge Density . 70 CHAPTER 8. Future Research Directions . 72 8.1 Computational Directions . 72 8.2 Theoretical Directions . 72 8.3 Tangential Directions . 74 REFERENCES . 76 APPENDIX A. SAGE Code . 77 A.1 Helper Functions . 77 A.2 allInducedPaths(G,u,v) . 78 A.3 MRV(G,A) . 80 A.4 computePks(G) . 82 A.5 computePksSparse(G) . 83 A.6 computePksSparseCustom(G,STACK) . 87 A.7 PksDictionary and related functions . 88 A.8 estimatePks(G,m) . 89 A.9 PansophyVariance(G) . 90 v ACKNOWLEDGMENTS Much thanks goes to my advisor Dr. Steve Butler. His passion for helping students learn is always an inspiration, and navigating the world of graduate mathematics was made easier with his help. Additionally, the chapter on density (and the fascinating results within) would not exist had he not suggested looking in that direction. Thanks to the ISU math department staff and faculty, who have supported me and dozens of other graduate students. We appreciate the hard work you do for us. Thanks to both my graduate and undergraduate friends, including but not limited to Beth, O'Neill, Adam, Michael, Joey, Peter and Raj. The time spent discussing math and hanging out was time well spent. I'm still not certain where I'm going after this, but I hope to keep in touch! And of course, a big thanks to Dr. Lazaros Kikas and Dr. Jeffery Boats, who introduced me to pansophy. This thesis wouldn't exist if I had missed their talk at the MIGHTY LVIII conference. vi ABSTRACT In 2017, Boats and Kikas introduced a new parameter, the pansophy of a graph, which is the expected value of the number of disjointly-routable paths in a graph given a random distribution of ordered starting and stopping points. We present an introduction to the topic and prove several results related to pansophy, including formulas and bounds for the pansophies of different graphs and graph families, the effect of various graph operations, and some density results. We also discuss the computational aspect of computing the pansophy of a graph. 1 CHAPTER 1. Introduction We begin by discussing the history of pansophy, namely the motivation and background leading to it becoming a studied graph invariant. For graph theory terminology not defined in this paper, refer to Diestel [2]. Unless otherwise stated, all graphs are finite. 1.1 Motivation and Background The origins of pansophy is rooted in the k-disjoint paths problem, an NP-complete problem from computer science: given k pairs of distinct vertices fs1; t1g to fsk; tkg, is it possible to connect the pairs together using vertex-disjoint paths? In 1992, Jwo et al. introduced the alternating group s1 t graph AGn, which is the Cayley graph where the vertices 1 are permutations in the alternating group An and where two vertices are adjacent if one gets from one permutation to the other via a rotation of symbols in the first, second and k-th position, where k ≥ 3. It was shown by Kikas, Cheng and Kruk that if n ≥ 5, t2 t1 then any n − 2 pairs of distinct vertices in AGn can be connected together using vertex-disjoint paths. However, Figure 1.1: The graph AG4. the result does not hold for n = 4, because there exists 2 pairs of vertices fs1; t1g and fs2; t2g such that routing with disjoint paths is impossible. This is illustrated in Figure 1.1. However, one of these authors, Lazaros Kikas, noticed that if two pairs of vertices were to 2 be selected at random, then there is a high probability that the vertex pairs can be successfully routed using vertex-disjoint paths. This leads to the question of not how many disjoint paths can be guaranteed, but instead how many disjoint paths can be expected. This expectation represents the new graph parameter Ψ(G), the pansophy of G, which was introduced by Lazaros Kikas and Jeffery Boats in 2017, and is the focus of this dissertation. The name, which in ancient Greek means \universal knowledge", is inspired by the NP-complete nature of the question { given a graph G and an algorithm for adaptively finding disjoint paths for a given random selection of vertex pairs, the pansophy of G represents the expected optimal performance, but a routing algorithm can accidentally block future connections if it chooses a poor path for the current vertex pair. To obtain the maximal routing volume possible, such an algorithm would need to be prescient enough to prevent this dilemma. 1.2 Definitions Definition 1.2.1. An unordered assignment A on graph G is a set containing pairwise-disjoint pairs of distinct vertices from G. We call A an unordered k-assignment if jAj = k. An ordered assignment is an ordered set containing pairwise-disjoint pairs of distinct vertices. We let V (A) denote the set of vertices contained in some pair in A, and we say A is an assignment on vertex subset S if V (A) ⊆ S. If it does not matter if A is ordered or not, then we will simply say \assignment" and \k- assignment". Also, note that jV (Ak)j = 2k holds for any k-assignment A. Remark 1.2.2. For every unordered k-assignment there are k! ordered k-assignments. Indeed, since there are k! different ways to permute k objects. n (2k)! Remark 1.2.3. There are 2k 2k ordered k-assignments on any n-vertex graph, n (2k)! n and so there are 2k 2kk! = 2k (2k − 1)!! unordered k-assignments. The first statement follows from a basic count: after choosing the 2k vertices in an ordered assignment A, we choose the pairs by permuting the order of the 2k vertices and then pairing the 3 n first vertex with the second, the third vertex with the fourth, and so on. There are 2k (2k)! ways to do this. However, the order within each pair doesn't matter, so we've overcounted by a factor k n (2k)! of 2 , hence there are 2k 2k ordered assignments. The second statement follows from the first statement and the previous remark. Definition 1.2.4. A k-assignment A is solvable if it can be disjointly-routed { that is, there exists k vertex-disjoint paths ρ1; ρ2; : : : ; ρk such that ρi is an si; ti-path for each fsi; tig 2 A.
Load more

Recommended publications
  • Math.GM] 1 Jul 2015 Itnegahwt Hoai Ubrfu Icvrdb Bro from Follows by Four Discovered of 1961, Four Bound in Number Lower Moser the Chromatic 7
    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]
  • 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]
  • Class Eight: Infinite Graphs
    Class Eight: Infinite Graphs Nothing in the definition of a graph prevented us from considering graphs with an infinite number of vertices. Most of the concepts and ideas we exam- ined with respect to finite graphs have an interesting infinite analogue. We are most interested in graph with vertex set N = f1; 2; 3; :::g and refer to such graphs as countable. As before we define the degree of a vertex to be the of vertices adjacent to it, but now we allow for the possibility that there are an infinite such neighbors. If every vertex in a graph on N has finite degree, then we call the graph locally finite. So that we stay in the world of graph theory we mostly examine countable graphs which are locally finite. Lastly, before we begin we recall two fundamental ideas from set theory. • Removing finitely many points from an infinite set yields an infinite set. This is sometimes know as the infinite pigeonhole principle. Think why... • A finite or countable union of countable sets is countable. This implies that a countable graph has a countable number of edges. Draw some \small" graphs and think about the following questions: How many edges can a countable graphs have? Describe all countable graphs such that every vertex has degree one. Describe all countable graphs such that every vertex has degree two. Describe a countable graph such that every vertex has either degree one or three. Give a countable graph with a vertex of every degree. Give a countable graph with a vertex of every degree and no cycles.
    [Show full text]
  • Proceedings of the 17Th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
    Proceedings of the 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimization Editors: Johann Hurink Stefan Klootwijk Bodo Manthey Victor Reijnders Martijn Schoot Uiterkamp Enschede, Netherlands, July 1{3, 2019 Editors Johann Hurink Stefan Klootwijk Bodo Manthey Victor Reijnders Martijn Schoot Uiterkamp CTW 2019 Proceedings of the 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimiza- tion J.L. Hurink, S. Klootwijk, B. Manthey, V.M.J.J. Reijnders, M.H.H. Schoot Uiterkamp (eds.) Enschede, University of Twente, Faculty of Electrical Engineering, Mathematics and Computer Science 1{3 July 2019 ISSN 2590-0870 DSI Workshop Proceedings Series (online) WP19-01 https://www.utwente.nl/en/digital-society/ c Copyright 2019; University of Twente, Enschede, Netherlands 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2019) CTW 2019 takes place at the University of Twente, Enschede, Netherlands, from July 1 to July 3, 2019. This volume collects the extended abstracts of the contributions that have been selected for presentation at the workshop. As it was the case with previous CTWs, we will edit a special edition of Discrete Applied Mathematics for CTW 2019. Hereby, we invite all participants to submit full-length papers related to the topics of the workshop. Program Committee: Ali F. Alkaya (Marmara University, Istanbul, Turkey) • Alberto Ceselli (Universit`adegli Studi di Milano, Italy) • Roberto Cordone (Universit`adegli Studi di Milano, Italy) • Ekrem Duman (Ozye˘ginUniversity,¨ Istanbul, Turkey) •
    [Show full text]
  • Generalized Petersen Graphs Are Unit-Distance Graphs 1
    J. Korean Math. Soc. 49 (2012), No. 3, pp. 475{491 http://dx.doi.org/10.4134/JKMS.2012.49.3.475 ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS Arjana Zitnik,ˇ Boris Horvat, and Tomaˇz Pisanski Abstract. In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n; j; k) admits a unit-distance rep- resentation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every I-graph I(n; j; k) has an isomor- phic I-graph that admits a unit-distance representation in the Euclidean plane with a n-fold rotational symmetry, with the exception of the fam- ilies I(n; j; j) and I(12m; m; 5m), m ≥ 1. We also provide unit-distance representations for these graphs. 1. Introduction I-graphs were introduced in the Foster census [5] and form a natural gen- eralization of the generalized Petersen graphs introduced by Coxeter [8] and named by Watkins [26]. This well-known family of graphs has been extensively studied [1, 10, 18, 20, 22, 25]. Let n ≥ 3 and j; k be such that 1 ≤ j, k < n and j; k 6= n=2. The I- graph I(n; j; k) is a graph with vertex set V (I(n; j; k)) = fu0; u1; : : : ; un−1; v0; v1; : : : ; vn−1g and edge set E(I(n; j; k)) = fuiui+j; uivi; vivi+k; i = 0; : : : ; n − 1g; where the subscripts are to be read modulo n.
    [Show full text]