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1 Lecture To 428 students: I did not intend to post these lecture notes, so please forgive their informal nature. They do not include pictures, but they cover all the material we went over in the rst class. 1. Graphs: Basic de nitions and examples. a). Informal, visual de nition by way of pictures. Informally, a graph is a nite set of points, called vertices, in the plane, and arcs, called edges, connecting vertices. Draw some pictures. Note: 1. Isolated vertices, vertices not connected to any others by edges are allowed; multiple edges between vertices are edges, called loops, from a vertex back to itself are also not allowed for the object we call graphs. A nite set of points with possibly multiple edges connecting them, or edges, called loops, which start and end at the same vertex, are called multigraphs. Terminology is not consistent and some authors use the term graph for multigraph and the term simple graph for what we call here a graph. For some also, multigraphs do not have loops and graphs with loops are called pseudographs. Moral: if you are reading another source, check carefully its basic de nitions. 2. In a drawing of a graph, edges may intersect at points that are not vertices; indeed, it is often not possible to draw a given graph in the plane so that all edges intersect only at vertex points. In drawing, indicate vertices by large dots so that intersections of edges not corresponding to vertices are not confused with vertices. b). Why are graphs interesting to study? Graphs occur as natural mathematical models for many problems in the natural and social sciences, as well as in Operations Research (managing complicated business and industrial procedures) and Computer Science. We give an example, called the marriage problem: Consider a group of n men and n women constituting a pool of single people wanting to get married . Every person wants to get married (single spouses only!), but only to a spouse they are comfortable with. Under what conditions can every person be matched with a consenting spouse, and how might one nd this compatible matching? We will model (but not solve) this problem using a graph. The idea is 1 to use a graph to represent the basic information in the problem, which is the set of compatible couples. We draw n vertices on the left, one per man, labeled from 1 to n (man 1, 2, etc.) We do the same for the women in a line on the right. We draw an edge between a vertex on the left and a vertex on the right if and only if the corresponding man and woman would consent to marrying one another. (Note: we are excluding from consideration the problem in which the men and women also have preferences among the partners they would accept.) The solution to the marriage problem is what is called a complete matching drawn from this edge set. This is a set a subset of edges of the marriage graph that , a subset that connects each man to a unique woman and vice-versa. This does not sound like a real-life problem, but it has real-life versions. One of the more prominent is the problem of matching medical school graduates with residencies. Here we give two n = 4 examples, one of which admits a matching, the other of which does not. c). Formal de nition of a graph. The mathematical de nition of a graph is an abstraction that, on the face of it, has nothing to do with the diagrams we have been drawing; rather, it formalizes the notion of a relation between elements of a nite set. Given a set G, an unordered pair of elements of G is a set of the form {x, y}, x and y being elements of G, possibly the same. Since {x, y} and {y, x} denote the same thing, no order of the elements x and y is implied. This notation is a little ambiguous, because we should really write {x, x} as {x} to be proper, but we will allow this informality. De nition: A multigraph G consists of a nite set V (G) of elements called vertices, and a nite list, or family, E(G) of unordered pairs of vertices of V (G). The members of E(G) are called edges. The word list or family is used instead of set for E(G) because an unordered pair can occur multiple times in E(g). (Some authors call such an E(G) a multi-set.) It is not demanded that the vertices of a pair are distinct; hence {u, u} is an acceptable edge. V (G) is called the vertex set of G; E(G) is called the vertex family of G. For simplicity, we shall use the notation xy for the edge {x, y}, keeping in mind that no order is implied. When the 2 edge set E(G) does not admit pairs of the form xy, nor multiple copies of a pair xy, then G is a graph proper. Examples 1. (Multigraph) (This example is di erent from the one in class which was an example of a graph.)V (G1 ) = {u, v, w, x}, E(G1 ) = {xx, xx, wx, wx, wv}. 2. V (G0 ) is the set of n men and n women of the marriage problem, E(G0 ) is the set of compatible couples. 3. The pictorial graphs we introduced above are interpreted as graphs in the sense of the rigorous de nition. by taking the vertex set to be the set of vertex points drawn in the plane and the edge family to be those unordered pairs, one per edge, connected by the edges of the pictorial graph. Write down the vertex and edge sets from one of the examples on the board. For this lecture, we will call graphs de ned in terms of vertices and edges drawn in the plane pictorial graphs or multigraphs, as the case may be, to distinguish them from graphs in abstract the sense. However, we shall not distinguish between them in a practical sense as we go through this course, because, just as any pictorial graph can be interpreted as a graph in the abstract sense, so can any abstract graph be represented by a pictorial graph. Given an abstract graph, draw a point in the plane corresponding to each element of a vertex set and draw an edge or loop between the graphical vertices for every unordered pair in the edge set. Illustrate on example 1. Observe this procedure is essentially what we did in discussing the marriage problem Note: The pictorial representation of an abstract graph is not unique, for if we use a di erent con guration of points in the plane to represent the vertices of the abstract graph we will get a di erent looking graph. But in some sense these pictorial graphs are all essentially same. We will formalize this sameness using the concept of graph isomorphism in the next lecture. 2. Elementary notions In this section we introduce basic concepts for discussing graphs. These concepts will start to reveal some of the issues graph theory is concerned 3 with. a). Order and Size. The order of a multigraph is the number of its vertices. The size of a multigraph is the number of its edges. b). Adjancey and incidence. Two edges are adjacent if they share a common vertex. Two vertices are adjacent if they share a common edge, that is if the edge between them is in the graph. An edge is incident to a vertex if that vertex is one of its endpoints. c). The degree of a vertex is the number of non-loop edges incident to that vertex, plus twice the number of loops at that vertex. We may think of the degree of a vertex as the number of lines exiting the vertex. Notation for degree of vertex v: d(v), or dG (v) if we want to make explicit the (multi)graph G. Question: What is the maximum size of a graph of order n? This would be the graph in which every vertex is connected to every other vertex. This is called the complete graph on n vertices. We will encounter it often. It is denoted Kn . Hence the edge set is the set of all unordered pairs of vertices {x, y}, x = y. This is the same as the set of all subsets of size 2 of the vertex set. There are n such subsets. 2 Question: What is the maximum size of a multigraph of order n? (Answer: There is no maximum size; we can add as many edges as we like to a multigraph.) d). Class Theorem 1. The sum of the degrees of all the vertices of a graph equals twice the number of its edges. In mathematical notation, dG (v) = |E(G)|. v G Proof: Given a vertex v, place a mark on each edge connecting v to another vertex. Please two marks on each loop from v to v. Do this for all vertices. The total number of marks will be v G dG (v). Each loop receives two marks. Each non-loop edge also receives two marks, one for each of its two vertices. Thus every edge has two marks, and so the total number of marks equals twice the number of edges. 4 This theorem is called the rst theorem of graph theory by some authors; also, it is called the handshake lemma. Imagine a party attended by n persons, each represented by a vertex. Represent each handshake by an edge between appropriate vertices. The degree of a vertex is the number of handshakes the person represented by that vertex participates in. Since each handshake involves two persons, summing up the degrees of all persons yields twice the number of handshakes total. Corollary. In any graph, the number of vertices of odd degree is even. Let G be a graph of order n. The list (dG (v1 ), dG (v2 ), . . . , dG (vn )) of the degrees of the vertices of G in increasing order is called the degree sequence of G. (Increasing means non-decreasing). Illustrate on some of the graphs we have drawn. The previous corollary implies that the terms of the degree sequence of a graph must add up to an even number. Thus not every sequence of n increasing numbers is the degree sequence of a graph. Example: (Taken from West, Intro to Graph Theory) Consider a league with two divisions and 13 teams in each division. Is it possible to schedule a season in which each team plays 9 games within its division and 4 games with a team from the other division? Our answer to this is no. Consider just trying to schedule the games for division I. We represent this as a graph with 13 vertices, one for each team. An edge between teams then represents a scheduled game between the teams. Consider just the graph for games between teams of division 1. To fulfull the requirements, each team would have to play 9 games intradivision. This means that each vertex would have to have degree 9. But by the Corollary we cannot have a graph of 13 vertices each of which has the odd degree 9. We can ask; if (d1 , . . . , dn ) is an increasing sequence of n nonnegative integers that total to an even number, is there a multigraph having that sequence as its degree sequence? The anwer is yes. Discussion Exercise; show this. It is more interesting to ask the same question of a (simple) graph. Of course, the answer is no because the number of edges of a 5 simple graph on n vertices is bounded by n . Also no vertex can 2 have degree higher than n 1. We could then ask if subject to these restriction and the restriction that the degrees sum to an even number, any sequence of n numbers is the degree sequence of a simple graph. The answer is still no. Exercise: construct an example. 6

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