THE LARGEST BOND IN 3-CONNECTED GRAPHS by Melissa Flynn A thesis submitted to the faculty of The University of Mississippi in partial fulfillment of the requirements of the Sally McDonnell Barksdale Honors College. Oxford May 2017 Approved by Advisor: Dr. Haidong Wu, Ph.D. Reader: Dr. Laura Sheppardson, Ph.D. Reader: Dr. Richard Gordon, Ph.D. © 2017 Melissa Flynn ALL RIGHTS RESERVED ii Abstract AgraphG is connected if given any two vertices, there is a path between them. A bond B is a minimal edge set in G such that G B has more components than − G. We say that a connected graph is dual Hamiltonian if its largest bond has size E(G) V (G) +2. In this thesis we verify the conjecture that any simple 3-connected | |−| | graph G has a largest bond with size at least ⌦(nlog32)(Ding,Dziobiak,Wu,2015 [3]) for a variety of graph classes including planar graphs, complete graphs, ladders, M¨obius ladders and circular ladders, complete bipartite graphs, some unique (3,g)- cages, the generalized Petersen graph, and some small hypercubes. We will also go further to prove that a variety of these graph classes not only satisfy the conjecture, but are also dual Hamiltonian. iii Contents List of Figures v 1 Introduction and Some Background on Graph Theory 1 1.1 Introduction . 1 1.2 Whatisagraph? ............................. 2 1.3 Cycles and Subgraphs . 3 1.4 Graph Connectivity . 4 1.5 Trees . 5 1.6 Planar Graphs . 6 1.6.1 The Planar Dual Graph . 7 1.7 DualHamiltonianGraphs ........................ 8 2 The Main Problem 9 2.1 The Conjecture . 9 2.2 Some Known Results . 9 2.2.1 The Lower Bound on Bond Size in a 3-connected Graph . 9 2.2.2 The Largest Cycle in a Planar Graph . 10 2.2.3 ResultsAboutDualHamiltonianGraphs . 10 3 Main Results 11 3.1 Introduction . 11 3.2 Planar Graphs . 11 3.2.1 Complete Graphs . 12 3.3 DualHamiltonianGraphs . .. 12 3.3.1 Ladders, M¨obiusLadders, and Circular Ladders . 12 3.3.2 The Complete Bipartite Graph . 15 3.3.3 The Generalized Petersen Graph . 17 3.3.4 Unique (3,g)-Cages . 19 3.4 AConjectureonHypercubes. 24 4 Further Study 28 Bibliography 29 iv List of Figures 1ExampleofaGraph...........................2 2AGraphContainingCycles.......................3 3Subgraphs.................................4 4 AnExampleofanEdge-Cut....................... 4 5 AnExampleofaBond.......................... 5 6 AnExampleofaTree .......................... 6 7AFewPlanarGraphs...........................6 8 Non-PlanarGraphs............................ 6 9TwoPossibleDrawingsofaPlanarGraph...............7 10 AGraphanditsDualGraph....................... 7 11 K3,K4,K5 ................................. 12 12 A Ladder Graph . 13 13 AMaximumBondinaLadderGraph. 13 14 TheM¨obiusLadder............................ 13 15 A Maximum Bond in the M¨obiusLadder . 14 16 A Circular Ladder . 14 17 AMaximumBondinaCircularLadder. 15 18 Examples of Bipartite Graphs . 16 19 A Maximum Bond in K3,3 ........................ 16 20 The Petersen Graph . 17 21 P (7, 3)................................... 17 22 A Maximum Bond in the Petersen Graph (P (5, 2)) . 19 23 A Maximum Bond in P (7, 3) . 19 24 A Maximum Bond on the (3,6)-Cage . 20 25 A Maximum Bond on the (3,7)-Cage . 21 26 A Maximum Bond on the (3,8)-Cage . 22 27 A Maximum Bond on the (3,11)-Cage . 23 28 The 2-cube . 24 29 The 3-cube . 24 30 A Maximum Bond in the 2-Cube . 25 31 A Maximum Bond in the 3-Cube . 25 32 A Maximum Bond in the 4-Cube . 26 33 A Maximum Bond in the 5-Cube . 27 v Chapter 1 Introduction and Some Background on Graph Theory 1.1 Introduction Graph Theory is a relatively young area of study in mathematics. The first doc- umented graph theory problem was the K¨onigsberg Bridge problem proposed by Leonhard Euler in 1736, long before the term “Graph Theory” was coined. This problem involved seven bridges connecting two islands to the main city of K¨onigsberg in Prussia, and the question was to devise a path which crossed each bridge only once. Ultimately, it was proven impossible to obtain such a path. It wasn’t until 1936 that the first textbook on graph theory was published. There are many practical applications for graph theory, typically involving relational modeling in areas such as biology, business and computer science. Graphs have many real-world applications such as modeling computer network systems and creating mappings for airline routes. In this thesis, we will specifically study bond sizes in graphs. Abondisaminimaledge-cut(aminimalsetofedgeswhosedeletiondisconnectsa connected graph). We know an upper bound for bond size in a graph is E(G) V (G) +2,where E(G) is the number of edges in the graph and V (G|) is the|− number| | of vertices in| the graph.| Additionally, we know that a 3-connected| graph| will 2 have a bond of size at least 17 plogn [3]; however, in 2015, Ding, Dziobiak, and Wu [3] raised the following conjecture: Conjecture (Ding, Dziobiak, Wu, 2015 [3]). Any simple 3-connected graph G will have a largest bond with size at least ⌦(nlog32) where n = V (G) . | | We say that f(n)=⌦(g(n)) if and only if there exists some constant M>0and some N N,suchthatf(n) Mg(n)foralln N. 2 ≥ ≥ 1 In this thesis, we will verify this conjecture for a variety of graph classes including planar graphs, complete graphs, ladders, M¨obius ladders and circular ladders, com- plete bipartite graphs, the generalized Petersen graph, a few unique (3,g)-cages and some small hypercubes. Furthermore, we will prove that a variety of these graph classes possess bonds which meet the maximum bound for bond size. These graphs are called dual Hamiltonian. We will show that graphs that meet this upper bound for bond size will verify our conjecture on the lower bound. 1.2 What is a graph? A simple graph G with n vertices and m edges consists of a vertex set V (G)= v1,v2,...,vn and an edge set E(G)= e1,e2,...,em ,whereeachedgeisanunordered pair{ of vertices} from V (G). We say that{ G =(V (G}),E(G)). A simple graph has no multiple edges or loops. A graph in which edges are allowed to repeat is called a multigraph. Unless otherwise stated, the graphs in this thesis are all simple graphs. The cardinality of a set S is the number of elements in the set, and is denoted S . Subsequently, the number of vertices in a graph (also called the order of the graph)| | can be denoted V (G) and the number of edges, E(G) .Most(butnotall)graphs have many visual| representations,| often called embeddings| | . To illustrate the above definitions, let’s use the following graph: G = A, B, C, D, E , A, B , B,C , C, D , C, E , D, E , D, A {{ } {{ } { } { } { } { } { }}} The edge set of G, E(G)is A, B , B,C , C, D , C, E , D, E , D, A . The vertex set of G, V (G)is{{ A, B,} C,{ D, E} {. } { } { } { }} V (G) =5, E(G) =6 { } Two| possible| | embeddings| of G are illustrated in Figure 1: Figure 1: Example of a Graph 2 We may abbreviate the edge a, b as ab.Wesaythata and b are called neighbors when they are joined by an edge.{ } Any two neighbors are said to be adjacent. When a and b are neighbors, we say that a and b are each incident to the edge ab.Subse- quently, the neighborhood of a vertex v,denotedN(v)isthesetofallneighborsofv. Using the example from Figure 1, we can see that N(A)= B,D .Thesizeofthe neighborhood of a vertex, N(v) ,iscalledthedegree of v,frequentlydenoted{ } d(v). Agraphinwhichallverticeshavedegree| | r is called r-regular. Many of the graphs we will discuss are cubic,meaningtheyare3-regular. The number of edges in a graph can be determined from the number of vertices and their degree by the following formula: The Handshake Lemma. 2 E(G) = v V (G) d(v). | | 2 P 1.3 Cycles and Subgraphs A walk is an alternating sequence of vertices and incident edges. In other words, a walk is a route that can be traveled within a graph from vertex to vertex along edges. A trail is a walk in which no edges are repeated, and a path is a walk in which no vertices are repeated. A walk is said to be closed if it begins and ends with the same vertex. A closed walk that repeats no vertices or edges is called a cycle. Figure 2: A Graph Containing Cycles In Figure 2, A, B, C, D, A is an example of a cycle. It follows a path from vertex A to B, C, D{ and back to} A to complete the cycle. Another example of a cycle is D, C, E, D . We call the size of the largest cycle in a graph the circumference of the graph,{ denoted} c(G), and the size of the smallest cycle in a graph the girth. AcycleonagraphG is called Hamiltonian if it contains every vertex from V (G). A subgraph of a graph G is a graph whose vertex set is a subset of V (G)andwhose edge set is a subset of E(G). We call a subgraph induced if it contains all edges which have both ends in the vertex set of the subgraph. The subgraph induced by the set S is denoted G[S]. 3 Figure 3: Subgraphs In Figure 3, both the graphs in the center and on the right are subgraphs of the leftmost graph. However, only the subgraph on the right is induced, as the edge AD has both ends in the vertex set of the subgraphs and thus, is required in the central subgraph in order for it to be induced.