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MMNA GRAPHS on EIGHT VERTICES OR FEWER a Thesis Presented To MMNA GRAPHS ON EIGHT VERTICES OR FEWER A Thesis Presented to the Faculty of California State University, Chico In Partial Fulfillment of the Requirements for the DeGree Master of Science in Mathematics Education by HuGo E. Ayala Summer 2014 TABLE OF CONTENTS PAGE List of Tables………………................................................................................................................. iv List of Figures..……………................................................................................................................. v Abstract.……………………................................................................................................................... xxi CHAPTER I. Introduction.………………………………………………………………………….. 1 Statement of Problem.…………………………………………………. 1 Map of Thesis..……………………………………………………………... 4 II. Definitions and Preliminary Constructions.…………………………….. 5 Definition of Terms.………………………………………….…..……... 5 Non-planar (7, 12) graphs..………………………………………….. 13 TrianGulations on Seven Vertices.………………………………… 35 Petersen Graphs and their Complements.……………………... 57 III. Proof of Theorem A.……………………………………………………………….. 72 Lemmas and Prior Results.…………………………………………... 72 Theorem 1.………………………………………………………………….. 75 Theorem 2.………………………………………………………………….. 84 References Cited.………………………………………………………………………………………….. 187 iii LIST OF TABLES TABLE PAGE 2.3.1 The Five Triangulations…………………………………………………………… 52 3.3.1 The Candidate MMNA Graphs…………………………………………………... 86 iv LIST OF FIGURES FIGURE PAGE 1.1.1 The Utility Problem.…………………………………………………………………. 1 1.1.2 The K3,3 Graph……………………..…………………………………………………… 2 1.1.3 The K5 Graph………...…………………………………………………………………. 2 1.1.4 The J1 Graph, the K6 Graph, and the K3,3,1 Graph………………………… 3 1.1.5 The P7 Graph, the K4,4 – e Graph, and the P8 Graph…………..…………. 4 2.1.1 Graph G………..……………………..…………………………………………………… 9 2.1.2 Graph G’..……..……………………..……………………………………………………. 9 2.1.3 Graph H...……..……………………..……………………………………………………. 10 2.1.4 Graph H with Vertex B Split…...………………………………………………….. 10 2.1.5 Graph J.....……..……………………..……………………………………………………. 11 2.1.6 Graph J with a (1,2) Split of Vertex B…………………………………………. 11 2.1.7 Graph J with a Second (1,2) Split of Vertex B…………………...…………. 12 2.1.8 Graph J with a Third (1,2) Split of Vertex B.……………………….………. 12 2.1.9 A DeGree Four Vertex, B.……..……………………………………………………. 12 2.1.10 A DeGree Four Vertex, B, Split into (0,4).……………………………………. 13 2.2.1 The K3,3 Graph……….…………………………………………………………………. 14 2.2.2 The Only Non-Planar (6,10) Graph with δ(G) > 2.………………………. 14 2.2.3 First Derived Non-Planar (6,11) Graph……..……………………….………. 15 v 2.2.4 Second Derived Non-Planar (6,11) Graph.…………………………………. 16 2.2.5 The K5 Graph……………………..………………..……………………………………. 16 2.2.6 Non-Planar (6,11) Graph Derived from K5…………………………………. 17 2.2.7 ComparinG Two Non-Planar Graphs. ………………………………………… 18 2.2.8 Third Derived Non-Planar (6,11) Graph…...……………………….………. 18 2.2.9 The Only Non-Planar (7,10) Graph with δ(G) > 2……………………….. 19 2.2.10 First Non-Planar (7,11) Graph……………..……………………………………. 20 2.2.11 Second Non-Planar (7,11) Graph. ..……………………………………………. 21 2.2.12 Third Non-Planar (7,11) Graph. ..………………………………………………. 21 2.2.13 Fourth Non-Planar (7,11) Graph. ..……………………………….……………. 22 2.2.14 Fifth Non-Planar (7,11) Graph. ..…….…………………………………………. 22 2.2.15 First (5, 42, 33, 2) Graph...……..……………………………………………………. 23 2.2.16 Second (5, 42, 33, 2) Graph...…….…………………………………………………. 24 2.2.17 Third (5, 42, 33, 2) Graph...…...….…………………………………………………. 24 2.2.18 The (5, 4, 35) Graph………………………………….……………………….………. 25 2.2.19 First (43, 34) Graph……….……..……………………………………………………. 25 2.2.20 Second (43, 34) Graph…...……..……………………………………………………. 26 2.2.21 Third (43, 34) Graph……….…...……………………………………………………. 26 2.2.22 First (44, 32, 2) Graph…………………………..……………………………………. 27 2.2.23 Second (44, 32, 2) Graph…………..…………..……………………………………. 28 2.2.24 Third (44, 32, 2) Graph……..…………………..……………………………………. 28 2.2.25 Fourth (44, 32, 2) Graph…………...…………..……………………………………. 29 vi 2.2.26 Fourth (43, 34) Graph…………………………..……………………………………. 29 2.2.27 Fifth (43, 34) Graph……………………….……..……………………………………. 30 2.2.28 First (45, 22) Graph.…………………….……..……………………………...………. 31 2.2.29 Second (45, 22) Graph.…………………..……..……………………………………. 31 2.2.30 Third (45, 22) Graph..…………………….……..……………………………………. 32 2.2.31 Fourth (5, 42, 33, 2) Graph…………….……..……………………………………. 33 2.2.32 The (52, 34, 2) Graph.…………………….……..……………………………………. 34 2.2.33 Fifth (44, 32, 2) Graph..………………….……..……………………………………. 34 2.3.1 A (5, 15) Graph…….……………………….……..……………………………………. 37 2.3.2 A (52, 25) Graph…...……………………….……..……………………………………. 38 2.3.3 A (52, 4, 32, 22) Graph..………………….……..……………………………………. 38 2.3.4 The (52, 45) Triangulation.…………….……..……………………………………. 39 2.3.5 A (6, 16) Graph…….……………………….……..……………………………………. 39 2.3.6 A (6, 36) Graph…….……………………….……..……………………………………. 40 2.3.7 A (6, 44, 32) Graph….…………………………………………………………………. 40 2.3.8 The (6, 46) Graph is Not a TrianGulation.…………………………………… 41 2.3.9 A (5, 15) Graph……..…………………….……..……………………………...………. 41 2.3.10 A (5, 4, 24, 1) Graph…..…………………..……..……………………………………. 42 2.3.11 A (5, 43, 33) Graph…..…………………….……..……………………………………. 42 2.3.12 A (52, 43, 32) Graph……………………….……..……………………………………. 43 2.3.13 The (53, 43, 3) Triangulation..…….….……..……………………………………. 43 2.3.14 A (6, 16) Graph……….....………………….……..……………………………………. 44 vii 2.3.15 A (6, 36) Graph.…………………………….……..……………………………………. 44 2.3.16 A (6, 5, 42, 33) Graph.…………………….……..……………………………………. 45 2.3.17 The (6, 5, 44, 3) Graph is Not a TrianGulation.……………………………. 45 2.3.18 A (6, 36) Graph……………….…………….……..……………………………………. 46 2.3.19 The (62, 43, 32) Triangulation….…….……..……………………………………. 46 2.3.20 A (6, 36) Graph……..…………………….……..……………………………...………. 47 2.3.21 A (6, 5, 42, 33) Graph...…………………..……..……………………………………. 47 2.3.22 The (6, 52, 42, 32) Triangulation…….……..……………………………………. 48 2.3.23 A (5, 35, 2) Graph………………………….……..……………………………………. 48 2.3.24 A (54, 32, 2) Graph……………....…….….……..……………………………………. 49 2.3.25 The (54, 4, 32) Graph is Not a TrianGulation….……………………………. 49 2.3.26 A (6, 36) Graph..…………...……………….……..……………………………………. 50 2.3.27 The (6, 53, 33) Triangulation.……….……..………………..……………………. 50 2.3.28 A (6, 16) Graph…………………………………………...……………………………. 51 2.3.29 A (62, 25) Graph.…………….…………….……..……………………………………. 51 2.3.30 The (63, 34) Graph is Not a TrianGulation….……….………………………. 52 2.3.31 (52, 45) Graph and its Complement..……..……………………………………. 53 2.3.32 The “Pentagon – SeGment” Graph….……..……………………………………. 53 2.3.33 Graph and its Complement………………………....……………………………. 53 2.3.34 The “Thin – Y” Graph.……..…………….……..……………………………………. 54 2.3.35 (62, 43, 32) Graph and its Complement.....……………………………………. 54 2.3.36 The “House” Graph…………………….……..……………………………...………. 55 viii 2.3.37 (6, 52, 42, 32) Graph and its Complement..…..………………………………. 55 2.3.38 The “Fat – Y” Graph………………..…….……..……………………………………. 55 2.3.39 (6, 53, 33) Graph and its Complement…..……………………………………. 56 2.3.40 The “Hat” Graph………………....…….….……..……………………………………. 56 2.3.41 A ListinG of Complements of the Five Triangulations on Seven Vertices………………………………………………………………………………. 56 2.4.1 The K6 Graph, the K3,3,1 Graph, and the P7 Graph………...………………. 57 2.4.2 The K4,4 – e Graph, the P8 Graph, and the P9 Graph.….…………………. 57 2.4.3 The Petersen Graph.…………………………………...……………………………. 58 2.4.4 The Petersen Graph P8 and its Complement.…..…………………………. 59 2.4.5 The Petersen Graph K4,4 – e and its Complement...….…………………. 59 2.4.6 The Petersen Graph P7 and its Complement...…….………………………. 59 2.4.7 The Complement of the P7 Graph…..……..……………………………………. 60 2.4.8 The P7 Graph Plus a Degree Zero Vertex, X………………...………………. 60 2.4.9 The P7 Complement Plus a Vertex X of Full DeGree….…………………. 60 2.4.10 The Petersen Graph P7…………………………………………………..…………. 61 2.4.11 The First Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 61 2.4.12 The Second Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 62 2.4.13 The Third Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 62 2.4.14 The Fourth Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 63 ix 2.4.15 The Fifth Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 63 2.4.16 The Sixth Graph on EiGht Vertices Derived from the P7 Graph, and its Complement..…..……………………………………………...………. 64 2.4.17 The Petersen Graph K3,3,1 and its Complement…....….…………………. 65 2.4.18 The K3,3,1 Complement Plus a Vertex X of Full DeGree...………………. 65 2.4.19 The First Graph on EiGht Vertices Derived from the K3,3,1 Graph, and its Complement..…..……………………………………………...………. 66 2.4.20 The Second Graph on EiGht Vertices Derived from the K3,3,1 Graph, and its Complement..…..……………………………………………...………. 66 2.4.21 The Third Graph on EiGht Vertices Derived from the K3,3,1 Graph, and its Complement..…..……………………………………………...………. 67 2.4.22 The K6 Graph……………………..………………..…………………………………… 68 2.4.23 A Graph on Seven Vertices Derived from the K6 Graph…….………… 68 2.4.24 A Graph on EiGht Vertices Derived from the K6 Graph..……...………. 69 2.4.25 The Complement Derived from a Graph of EiGht Vertices that has a K6 Minor..…..……..…………………………….…………………………. 69 2.4.26 Complements Constructed from the Two Petersen Graphs of EiGht Vertices..…..………………..…………………………….…………………………. 70 2.4.27 Complements Created from the P7 Graph on EiGht Vertices……..…. 70 2.4.28 Complements Created from the K3,3,1 Graph on EiGht Vertices….…. 71 2.4.29 The Complement Created from the K6 Graph on EiGht Vertices..…. 71 3.1.1
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