Clemson University TigerPrints All Dissertations Dissertations 8-2007 Automorphic Decompositions of Graphs Robert Beeler Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Part of the Applied Mathematics Commons Recommended Citation Beeler, Robert, "Automorphic Decompositions of Graphs" (2007). All Dissertations. 92. https://tigerprints.clemson.edu/all_dissertations/92 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. AUTOMORPHIC DECOMPOSITIONS OF GRAPHS A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mathematical Sciences by Robert A. Beeler August 2007 Accepted by: Dr. Robert E. Jamison, Committee Chair Dr. Neil J. Calkin Dr. Gretchen L. Matthews Dr. Douglas R. Shier ABSTRACT Let G and H be graphs. A G-decomposition D of a graph H is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. It is well known that a graceful labelling (or more generally a ρ-valuation) of a graph G induces a cyclic G-decomposition of a complete graph. We will extend these notions to that of a general valuation in a cyclic group. Such valuations yield decompositions of circulant graphs. We will show that every graph has a valuation and hence is a divisor of arbitrarily large circulant graphs. The problem of which graphs can host a decomposition by a graph G is much more difficult and is still open. Let D be a G-decomposition of H. The intersection graph generated by the decom- position D, denoted I(D), has a vertex for each part of the partition and an edge if the parts of the partition share a common node in H. While the problem of determining whether a G-decomposition of H exists is a well-studied one, the structure of the resulting intersection graph has not been the subject of much research. As such, we are interested in the structure of these graphs. Of specific interest is whether there exist H, G, and a G-decomposition of H such that the resulting intersection graph is isomorphic to H. A decomposition D such that I(D) =∼ H is said to be automorphic (“self-shaped”). If a graph H is able to host an automorphic decomposition, then we say that H is an automorphic host. Similarly, if there exists an H such that there exists an automorphic G-decomposition of H, then we say that G is an automorphic divisor. In this dissertation, we will give several necessary conditions for the existence of an automorphic G-decomposition of H. These will allow us to examine the problem of which graphs are automorphic hosts. We conjecture that only even regular graphs are able to host an automorphic decomposition. In an attempt to prove this conjecture, we have shown the following special cases: iv • If χ(H) ≤ 3, then H is 2e(G)-regular. • If χ(H)= n(G) and G is d-regular, then H is n(G)d-regular. • If χ(H) = n(G), G is not a disjoint union of P2’s, and the smallest elements of dseq(G) are 1 and a, where a ≥ 2e(G) − n(G) + 1, then H must be 2e(G)-regular. • If G =∼ P4 and χ(H) = 4, then H is 6-regular. • If G is d-regular and any two blocks share at most one common node in H, then H is n(G)d-regular. This dissertation will also examine the problem of which graphs are automorphic divisors. We conjecture that every graph is an automorphic divisor. In an attempt to prove this, we generalize our notion of a valuation for arbitrary groups. This allows us to consider direct products of the related valuations. ACKNOWLEDGMENTS I would first like to thank my family, D. Beeler, L. Beeler, J. Beeler, and P. Keck for their love and support throughout my life. To M. Bennett for carefully proofreading the dissertation. To J. Chrispell for his assistance in formatting the document. To my advisor, R. Jamison for his guidance in constructing and editing this dissertation. To my committee members, N. Calkin, G. Matthews, and D. Shier, for their comments and suggestions in the final stages of this process. Finally, I wish to acknowledge the other excellent math teachers in my career. In particular, I would like to thank J. Dydak, R. Sharp, C. Wagner, J. Xiong, and especially D. Vinson. TABLE OF CONTENTS Page TITLEPAGE ....................................... i ABSTRACT ........................................ iii ACKNOWLEDGMENTS ................................. v LISTOFTABLES..................................... ix LISTOFFIGURES .................................... xi CHAPTER 1. BASICDEFINITIONS .............................. 1 1.1 BasicGraphTheory............................. 1 1.2 A Brief Introduction to Abstract Algebra and Number Theory..... 12 1.3 Factorizations and Matchings of Graphs . 14 1.4 CombinatorialDesigns ........................... 15 1.5 Preview.................................... 19 2. GRAPHDECOMPOSITIONS.......................... 23 2.1 Labellings and Valuations on Graphs . 24 2.2 Circulants .................................. 29 2.3 DecompositionsofGraphs ......................... 35 2.4 IntersectionGraphs ............................. 47 3. AUTOMORPHICDECOMPOSITIONS . 55 3.1 Examples of Automorphic Decompositions . 55 3.2 NecessaryConditions ............................ 61 4. AUTOMORPHICHOSTS ............................ 69 4.1 Simple Automorphic Decompositions . 69 4.2 Fully Automorphic Decompositions . 74 4.3 Automorphic Hosts with a Pendant Node . 82 5. AUTOMORPHICDIVISORS .......................... 93 5.1 Transulants, Translational Decompositions, and Γ-valuations . 93 5.2 Subproducts and Subjoins of Graphs . 99 5.3 OtherConstructions............................. 110 5.4 PersistentAutomorphicDivisors . 118 viii Table of Contents (Continued) Page 5.5 Fully Automorphic Divisors and Their Hosts . 123 6. EXTENSIONS................................... 129 6.1 Automorphic Decompositions of Multigraphs . 129 6.2 FamiliesofPrototypes ........................... 136 7. CONCLUDINGREMARKS ........................... 139 APPENDICES A. List of P3 Intersections Representing a Vertex of Degree Two . 143 B.ListofNotation .................................. 147 BIBLIOGRAPHY ..................................... 149 LIST OF TABLES Table Page 2.1 Lengths of Small Optimal Golomb Rulers . ..... 42 2.2 SmallOptimalGolombRulers. 43 2.3 SmallDifferenceSets.............................. 45 5.1 Examples of Fully Automorphic Decompositions . ........ 123 5.2 p-Modular Valuations of Kp for p ≤ 6 ...................... 126 LIST OF FIGURES Figure Page 1.1 AGraph....................................... 2 1.2 SpecialGraphsonSixVertices . .... 5 1.3 ProductsonGraphs ................................ 8 1.4 AGraphanditsLineGraph ........................... 9 1.5 An Automorphic P3-Decomposition of K5 .................... 11 1.6 ProjectivePlaneofOrder2 . 17 2.1 A Gracefully Labelled Caterpillar - C(6, 1, 4) .................. 27 2.2 Graceful Labellings For Non-Caterpillar Trees With n(G) ≤ 9......... 28 2.3 Graceful Labellings of a Non-Caterpillar Tree With n(G)=7 ......... 28 2.4 Circulant - C6(1, 2)................................. 29 2.5 A Z63-Valuation on K6 ............................... 41 2.6 Q3 and Intersection Graphs of P3-Decompositions. 48 2.7 All Graphs Can Be Represented As Decomposition Graphs . ........ 50 2.8 Graphs That Are Not Vertex Induced Subgraphs of Line Graphs ....... 51 3.1 K3 ∨ C6(2, 3) and a K3-DecompositionGraph . 60 4.1 Possible P3 Intersections Representing a Node of Degree One . 83 4.2 AnExposedNode ................................. 88 5.1 Γ-Transulant CD6 (r, sr) .............................. 94 5.2 A (Z3 × Z3)-ValuationofaGraph ........................ 95 5.3 Various Subproducts of P2 and P3 ........................ 101 5.4 Subjoins of P2 and P3 ............................... 107 5.5 Superimpositions of P2 and P3 .......................... 111 5.6 A Closed Z13-Valuation of the “Bowtie” . 120 6.1 An Extension of H ................................. 130 xii List of Figures (Continued) Figure Page 6.2 Multi-Circulant - C6((1, 1), (2, 2))......................... 131 6.3 An Automorphic P2-Decomposition of 2P2 .................... 134 6.4 An Automorphic K-Decomposition of K4 − e .................. 137 CHAPTER 1 BASIC DEFINITIONS The goal of this chapter is to introduce the major ideas that will be used throughout the dissertation. Readers who are familiar with the standard texts will recognize many of these terms. As notation often varies in different sources, we wish to standardize the internal notation of this document. 1.1 Basic Graph Theory A (simple) graph G is a triple consisting of a vertex set V (G), an edge set E(G), and an incidence relation that associates with each edge exactly two vertices called its endpoints. Often, we will use the more compact notation G = (V,E). In this case the incidence relation will be implicit. An edge e with endpoints u,v ∈ V (G) will often be denoted uv. If uv ∈ E(G), then we say that u and v are adjacent and are neighbors. We say that an edge e is incident with a vertex v if and only if e has v as one of its endpoints. A graph G = (V,E) is finite if and only if V and E are both finite sets [75]. Unless otherwise explicitly stated, we will assume that all graphs in this dissertation are finite. Let G and H be graphs. We say that G is a subgraph, or more properly, an edge induced subgraph, of H if V (G) ⊆ V (H) and E(G) ⊆ E(H). In this case, we say that H contains G and write G ⊆ H [75]. Let A ⊆ V (G). The subgraph of G induced by A, denoted GA, is the graph with vertex set A and E(GA) = {xy : x, y ∈ A, xy ∈ E(G)}. Such graphs are called vertex induced subgraphs [33]. An isomorphism between two graphs G and H is a bijection f : V (G) → V (H) such that uv ∈ E(G) if and only if f(u)f(v) ∈ E(H). If such an isomorphism exists, we say that G is isomorphic to H, and denote this by G =∼ H. We say that H is G-free if for all A ⊆ V (H), HA is not isomorphic to G.