Continuing Training Program for Mathematics Teachers inNorthern Luzon - Part IV

Saint Mary’s UniversityBayombong, Nueva Vizcaya

October 26-27, 2010

SELECTED TOPICS IN GRAPH THEORY

Phoebe Chloe F. RamosUniversity of the Philippines Baguio

Chapter 1

Introduction

1.1 Why Study Graphs?

The best way to illustrate the utility of graphs is via a ”Cook’s tour” of several simple problems thatcan be stated and solved via graph theory. We do this in an intuitive manner prior to presentingformal definitions. Graph theory has many practical applications in various disciplines, including,to name a few, biology, computer science, economics, engineering, informatics, linguistics, mathe-matics, medicine, and social science. As will become evident after reading this chapter, graphs areexcellent modeling tools. We now look at some several classic problems.

The Bridges of Konigsberg

Euler (1707- 1782) became the father of graph theory as well as topology when in 1736 hesettled a famous unsolved problem of his day called the Konigsberg Bridge Problem.

The river Pregel separates the city of Konigsberg into 4 separate regions and the regions areconnected by 7 bridges. In the summer evenings, the citizens of the country would like to havea walk around the whole city. Some curious citizens wondered whether it is possible to begin atone of the regions, cross each bridge exactly once and return to the same starting point. Can thecitizen’s suggestion be made possible?

Figure 1.1: The Seven Bridges of Konigsberg

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•... .Figure 1.2: The graph of the seven bridges of Konigsberg

The next problem we discuss concerns discovering communities on the World Wide Web (WWW).

World Wide Web Communities

The World Wide Web can be modeled as a graph, where the web pages are represented by dots orvertices and the hyperlinks between them are represented by lines or edges in the graph. One candiscover interesting information by examining this ”Web Graph”. As an example, the graph shownis termed a Web community. This name is bestowed because the vertices represent two differentclasses of objects, and each vertex representing one type of object is connected to every vertexrepresenting the other kind of objects.

Such web communities can be discovered by finding complete bipartite subgraphs in the Web graph.Web community information can be used for marketing purposes or for examining the relationshipsamong companies in a given industry.

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Job Assignments

A certain company has five different jobs labeled by Ji, where 1 ≤ i ≤ 5, and wants to hirepeople for these positions. The seven applicants, labeled by Aj for 1 ≤ j ≤ 7, are qualified forsome of the positions, but not for others. An edge between an applicant Aj and a job Ji meansthat applicant Aj is qualified for job Ji. This graph is an example of a bipartite graph, where thevertices represent two kinds of objects. The question in this problem is : Can the company fill all

its open positions with qualified applicants?

Figure 1.3: Bipartite graph describing job qualifications

Storing Volatile Chemicals

A particular chemical factory produces seven different kinds of chemicals, labeled as C1, C2, · · · , C7.For security reasons some of the chemicals should not be stored in the same warehouse. The graphdepicted shows the situation of volatility between the chemicals, where an edge between chemicalsCi and Cj indicates a grave danger in storing these chemicals in the same warehouse, whereas lackof an edge indicates it is safe to store the chemicals in the same warehouse. Here is the question:What is the minimum number of warehouse the factory needs in order to store its chemical productssafely?

Chapter 2

Basic Concepts in Graph Theory

2.1 The Definition of a Graph

A graph G consists of a finite nonempty set V (G) of objects called vertices together with a setE(G) of unordered pairs of vertices; the elements of E(G) are called edges. The sets V (G) andE(G) are called the vertex set and edge set of G respectively. So a graph G is a pair (actuallyan ordered pair) of two sets V (G) and E(G). For this reason, some write G = (V,E). Vertices aresometmes called points or nodes and edges are sometimes called lines. Two graphs G and H areequal if V (G) = V (H) and E(G) = E(H), in which case we write G = H.

It is common to represent a graph by a diagram in the plane where the vertices are repre-sented by points and whose edges are indicated by the presence of a line segment or curve betweenthe two points in the plane corresponding to the appropriate vertices. The diagram itself is referredto as a graph.

Figure 2.1: A Graph G

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For the graph G above, the vertex set V (G) and the edge set E(G) are

V (G) = {c1, c2, c3, c4, c5, c6, c7}E(G) = {{c1, c2} , {c1, c3} , {c1, c5} , {c1, c7} , {c2, c3} , {c2, c4} , {c2, c7} ,

{c3, c4} , {c3, c5} , {c4, c5} , {c4, c6} , {c4, c7} , {c6, c7}}

When dealing with graphs, it is customary and simpler to write a 2-element set or an edge{u, v} in G as uv (or vu). If uv is an edge of G, then u and v are said to be adjacent in G. Thenumber of vertices in G is often called the order of G while the number of edges is its size. Sincethe vertex set of every graph is nonempty, the order of every graph is at least 1. A graph withexactly one vertex is called a trivial graph, implying that the order of a nontrivial graph is atleast 2.

Figure 2.2: A trivial graph

The graph G in Figure 2.1 has order 7 and size 13. We often use n and m for the order and size,respectively, of a graph. So for the graph in Figure 1.3, n = 7 and m = 13.

A graph G with V (G) = {u, v, w, x, y} and E(G) = {uv, uw, vw, vx, wx, xy} is shown in Figure1.4. There are occasions when we are interested in the structure of the graph and not in what thevertices are called. In this case a graph is drawn without labeling its vertices. For this reason, thegraph G of Figure 2.3 (a) is a labeled graph and Figure 1.4 (b) represents an unlabeled graph.

Figure 2.3: A labeled graph and an unlabeled graph

Exercises

1. Let S = {2, 3, 4, 7, 11, 13}. Draw the graph G whose vertex set is S and such that ij ∈ E(G)for i, j ∈ S if i + j ∈ S or |i− j| ∈ S.

2. Suppose that we have a collection of 3-letter English words, say

ACT, AIM, ARC, ARM, ART, CAR, CAT, OAR, OAT, RAT, TAR.

We say that a word W1 can be transformed into a word W2 if W2 can be obtained from W1

by performing exactly one of the following two steps:

(a) interchanging two letters of W1;

(b) replacing a letter in W1 by another letter.

Therefore, if W1 can be transformed into W2, then W2 can be transformed into W1. Thissituation can be modeled by a graph G, where the given words are the vertices of G and twovertices are adjacent in G if the corresponding words can be transformed into each other.This graph is called the word graph of the set of words. Draw the graph G.

3. Give an example of five 3-letter words whose word graph is shown below.(with the verticesappropriately labeled).

Figure 2.4: The graph for exercise 3

4. Give a set of five 3-letter words whose word graph is shown below.(with the vertices appro-priately labeled)

Figure 2.5: The graph for exercise 4

5. The figure below illustrates the traffic lanes at the intersection of two streets. When a vehicleapproaches this intersection, it could be in one of the seven lanes: L1, L2, L3,L4,L5, L6, L7.Draw a graph G that models this situation, where V (G) = {L1, L2, L3, L4, L5, L6, L7} andwhere two vertices are joined by an edge if vehicles in these two lanes cannot safely enter thisintersection at the same time.

2.2 Subgraphs

In order to analyze certain situations that can be modeled by graphs, we must have a better un-derstanding of graphs. As with all areas of mathematics, there is a certain amount of terminology

Figure 2.6: Traffic lanes.

with which we must be familiar in order to discuss graphs and their properties. Learning thisfundamental terminology is our current goal.

Let us review some concepts and introduce others. If e = uv is an edge of G, then u and vare adjacent vertices. We also say u and v are joined by the edge e. The vertices u and v arereferred to as neighbors of each other. In this case, the vertex u and the edge e (as well as vand e) are said to be incident with each other. Distinct edges incident with a common vertex areadjacent edges.

A graph H is called a subgraph of a graph G, written H ⊂ G, if V (H) ⊂ V (G) andE(H) ⊂ E(G). We also say that G contains H as a subgraph. If H ⊂ G and either V (H) is aproper subset of V (G) or E(H) is a proper subset of E(G), then H is a proper subgraph of G.So the graph H of Figure 2.4 is a subgraph of the graph of G shown in the figure; indeed, H isa proper subgraph of G. If a subgraph of a graph G has the same vertex set as G, then it is aspanning subgraph of G.

A subgraph F of a graph G is called an induced subgraph of G if whenever u and v arevertices of F and uv is an edge of G, then uv is an edge of F as well.

If S is a nonempty set of vertices of a graph G, then the subgraph of G induced by Sis the induced subgraph with vertex set S. The induced subgraph is denoted by 〈S〉. To emphasizethat this is an induced subgraph of G, we sometimes denote the subgraph by 〈S〉G.

Figure 2.7: A graph G and some of its subgraphs

For a nonempty set X of edges , the subgraph 〈X〉 induced by X has edge set X andconsists of all vertices that are incident with at least one edge in X. This subgraph is called anedge-induced subgraph of G. Sometimes G[S] and G[X] are used for 〈S〉 and 〈X〉 , respectively.

Any proper subgraph of a graph G can be obtained by removing vertices and edges from G. Foran edge e of G, we write G− e for the spanning subgraph of G whose edge set consists of all edgesof G except e. More generally, if X is a set of edges of G, then G−X is the spanning subgraph of Gwith E(G−X) = E(G)−X. If X = {e1, e2, · · · , ek}, then we also write G−X as G−e1−e2−· · ·−ek.

For a vertex v of a nontrivial graph G, the subgraph G − v consists of all vertices of G ex-cept v and all edges of G except those incident with v. For a proper subset U of V (G), thesubgraph G − U has vertex set V (G) − U and its edge set consists of all edges of G joining twovertices in V (G)− U . Necessarily, G− U is an induced subgraph of G; indeed G− U = 〈V (G)〉.

If u and v are nonadjacent vertices of a graph G, then e = uv /∈ E(G). By G + e, we mean thegraph with vertex set V (G) and edge set E(G) ∪ {e}. Thus, G is a spanning subgraph of G + e.

2.3 Walks, Paths and Cycles

Many of the concepts that occur in graph theory and which we will investigate in detail later con-cern various ways in which one can ’move about’ in a graph.

Let’s start at some vertex u of a graph G. If we proceed from u to a neighbor of u and

then to a neighbor of that vertex, and so on, until we finally come to a stop at a vertex v, then wehave just described a walk from u to v in G. More formally, a u− v walk W in G is a sequence ofvertices in G, beginning with u and ending at v such that consecutive vertices in the sequence areadjacent (or they form an edge), that is, we can express W as

W : u = v0, v1, · · · , vk = v,

where k ≥ 0 and vi and vi+1 are adjacent for i = 0, 1, 2, · · · , k − 1. Each vertex vi (0 ≤ i ≤ k)and each edge vivi+1 (0 ≤ i ≤ k) is said to lie on or belong to W . Notice that the definition of thewalk W does not require the listed vertices to be distinct; in fact, even u and v are not required tobe distinct. However, every two consecutive vertices in W are distinct since they are adjacent. Ifu = v, then the walk is closed; while if u 6= v, then W is open.

The number of edges encountered in a walk (including multiple occurrences of an edge) is calledthe length of the walk.

For the graph G above,

Figure 2.8: Illustrating walks in a graph G

W : x, y, w, y, v, w

is an x− w walk of length 5 ( one less than the number of occurrences of vertices in the walk). Awalk of length 0 is a trivial walk. So

W : v

is a trivial walk.

We define a u − v trail in a graph G to be a u − v walk in which no edge is traversed morethan once. Thus, the x−w walk W is not an x−w trail as the edge wy is repeated. On the otherhand

T : u, w, y, x, w, v

is a u− v trail. Notice that this trail repeats the vertex w.

A u − v walk in a graph in which no vertices are repeated is a u − v path. While the u − vtrail T is not a u− v path,

P : u, w, y, v

is a u − v path. If no vertex in a walk is repeated (thereby producing a path), then no edge isrepeated either. Hence, every path is a trail.

If a u − v walk in a graph is followed by a v − w path, then a u − w walk results. In particu-lar, a u− v path followed by a v−w path is a u−w walk W , but not necessarily a u−w path, asvertices in W may be repeated. While not every walk is a path, if a graph contains a u − v walk,then it must also contain a u− v path.

Theorem 1. If a graph G contains a u− v walk of length l, then G contains a u− v path of lengthat most l.

A walk is closed when the first and the last vertices, x0 and xn are the same. Closed walks arealso called circuits. Sometimes, they also refer to a circuit in a graph G as a closed trail of length3 or more. Hence a circuit begins and ends at the same vertex but repeats no edge. For the graphin Figure 2.5, the following are circuits:

C : y, w, u, v, w, x, y

: x, y, w, u, v, w, x

: w, x, y, w, u, v, w

A cycle of length n is a closed walk of length n, n ≥ 3, in which no vertex is repeated. A k− cycleis a cycle of lenth k. A 3-cycle is also referred to as a triangle. A cycle of odd length is called anodd cycle; while a cycle of even length is called an even cycle. The following is an example of acycle, namely a 4-cycle.

C ′ : x, y, v, w, x

The vertices ad edges of a trail, path, cicuit, or cycle in a graph G form a subgraph of G, also calleda trail, path, circuit or cycle. Hence a path, for example is used to describe both a manner oftraversing certain vertices and edges of G and a subgraph consisting of those vertices and edges.The subgraphs G1, G2, G3, G4 of the graph G in Figure 2.5 are a trail, path, circuit, and cycle,respectively.We will have special interest in graphs G in which it is possible to travel from each vertex of G

to any other vertex of G. If G contains a u− v path, then u and v are said to be connected andu is connected to v (and v is connected to u). So, saying that u and v are connected only meansthat there is some u − v path in G; it doesn’t say that u and v are joined by an edge. Of course,if u is joined to v, then u is connected to v as well. A graph G is connected if every two ver-tices of G are connected, that is, if G contains a u−v path for every pair u, v of distinct vertices of G.

A graph that is not connected is called disconnected. A connected subgraph of G that is nota proper subgraph of any other connected subgraph of G is a component of G. A graph G isconnected if and only if it has exactly one component.

The graph G in Figure 2.5 is connected, while graph H below is disconnected since, there isno s−w path in H. There is no x− z path either.The graph H has three components namely H1,H2 and H3.

Figure 2.9: Trails, paths, circuits and cycles as subgraphs of a graph

Figure 2.10: A disconnected graph and its components

Let G be a connected graph of order n, and let u and v be two vertices of G. The distance betweenu and v is the smallest length of any u− v path in G and is denoted by dG(u, v) or simply d(u, v)if the graph G under consideration is clear. Hence, if d(u, v) = k, then there exists a u− v path

P : u = v0, v1, · · · , vk = v

of length k in G but no u − v path of smaller length exists in G. A u − v path of length d(u, v)is called a u − v geodesic. In fact, since the path P above is a u − v geodesic, not only isd(u, v) = d(u, vk) = k, but d(u, vi) = i for every i with 0 ≤ i ≤ k.

If u = v, then d(u, v) = 0. If uv ∈ E(G), then d(u, v) = 1. In general, 0 ≤ d(u, v) ≤ n − 1for every two vertices u and v (distinct or not) in a connected graph of order n. For the verticesu and y in the graph in Figure 2.5, d(u, y) = 2. If G is disconnected, then there are some pairsx, y of distinct vertices of G such that there is no x−y path in G. In this case, d(x, y) is not defined.

The greatest distance between any two vertices of a connected graph G is called the diam-eter of G and is denoted by diam(G).

Exercises

1. Let G be the graph shown below, let X = {e, f}, where e = ru and f = vw, and letU = {u, w}. Draw the subgraphs G−X and G− U of G.

Figure 2.11: The graph G for items 1-2

2. For the graph G of Figure 2.8, give an example of each of the following or explain why nosuch example exists.(a) An x− y walk of length 6.(b) A v − w trail that is not a v − w path.(c) An r − z path of length 2.(d) An x− z path of length 3.(e) An x− t path of length d(x, t).(f) A circuit of length 10.(g) A cycle of length 8.(h) A geodesic whose length is diam(G).

3. Prove that if P and Q are two longest paths in a connected graph, then P and Q have atleast one vertex in common.

4. Prove or disprove: Let G be a connected graph of diameter k. If P and Q are two geodesicsof length k in G, then P and Q have at least one vertex in common.

2.4 Common Classes of Graphs

As we continue to study graphs, we will see that there are certain graphs that are encountered oftenand it is useful to be familiar with them. In many instances, there is special notation reserved forthese graphs.

We have already seen that paths and cycles are certain kind of walks in graphs. These termsare also used to describe certain kinds of graphs. If the vertices of a graph G of order n can belabeled (or relabeled) v1, v2, · · · , vn so that its edges are v1v2, v2v3, · · · , vn−1vn, then G is called a

path; while if the vertices of a graph G of order n ≥ 3 can be labeled (or relabeled ) v1, v2, · · · , vn

so that its edges are v1v2, v2v3, · · · , vn−1vn, and v1vn, then G is called a cycle.

A graph that is a path of order n is denoted by Pn, while a graph that is a cycle of ordern ≥ 3 is denoted by Cn. Several paths and cycles are shown below.

Figure 2.12: Paths and cycles

A graph G is complete if every two distinct vertices of G are adjacent. A complete graph oforder n is denoted by Kn. Therefore, Kn has the maximum possible size for a graph with n vertices.Since every two distinct vertices of Kn are joined by an edge, the number of pairs of vertices in Kn

is(

n

2

)and so

the size of Kn is(

n

2

)=

n(n− 1)2

.

Therefore, the complete graph K3 has three edges, K4 has six edges, and K5 has ten edges. Thefirst five complete graphs are shown in Figure . Notice that P1 and K1 represent the same graph,as do P2 and K2, as well as C3 and K3. Although there are edges that cross in the drawings of K4

and K5, the points of intersection do not represent vertices.Although we have attempted to draw these graphs in a manner that makes them easy to visualize,

this is certainly not a requirement when drawing a graph, as its vertices can be placed in anyconvenient location. The figure below shows a variety of ways to draw the path P4 and completegraph K4.

The complement G of a graph G is that graph whose vertex set is V (G) and such that foreach pair u, v of vertices of G, uv is an edge of G if and only if uv is not an edge of G. Observe that

if G is a graph of order n and size m, then G is a graph of order n and size(

n

2

)−m. The graph

Kn then has n vertices and no edges; it is called the empty graph of order n. Therefore, emptygraphs have empty edge sets. In fact, if G is any graph of order n, then G − E(G) is the empty

Figure 2.13: Complete graphs

Figure 2.14: The graphs P4 and K4

graph Kn. By definition, no graph can have an empty vertex set. A graph H and its complementare shown below. Both of these graphs are connected. Although a graph and its complement neednot both be connected, at least one must be connected.

Theorem 2. If G is a disconnected graph, then G is connected.

We now turn to graphs whose vertex sets can be partitioned in special ways. A graph G is abipartite graph if V (G) can be partitioned into two subsets U and W , called partite sets, suchthat every edge of G joins a vertex of U and a vertex of W . It’s not always easy to tell at a glancewhether a graph is bipartite.

For example, the connected graphs G1 and G2 of Figure 2.13 are bipartite as every edge of G1

joins a vertex of U1 = {u1, x1, y1} and a vertex of W1 = {v1, w1}, while every edge of G2 joins avertex of U2 = {u2, w2, y2} and a vertex of W2 = {v2, x2, z2}. The bipartite nature of these graphsis illustrated in Figure 2.13.

Figure 2.15: A graph and its complement

Figure 2.16: Bipartite graphs

Theorem 3. A nontrivial graph G is a bipartite graph if and only if G contains no odd cycles.

We know that if G is a bipartite graph, then V (G) can be partitioned into two subsets U and W ,called partite sets, such that every edge of G joins a vertex of U and a vertex of W . However,this does not mean that every vertex of U is adjacent to every vertex of W . If this does happen,however, then we call G a complete bipartite graph. A complete bipartite graph with |U | = sand |W | = t is denoted by Ks,t or Kt,s. If either s = 1 or t = 1, then Ks,t is a star.

2.5 Multigraphs and Digraphs

We can obtain similar structures by altering our definition in various ways. Here are some examples.

1. By replacing our set E(G) with a set of ordered pairs of vertices, we obtain a directed graphor digraph . Each edge of a digraph has a specific orientation.

Figure 2.17: A digraph.

2. If we allow repeated elements in our set of edges, technically replacing our set E(G) with amultiset, we obtain a multigraph.

Figure 2.18: A multigraph

3. By allowing edges to connect a vertex to itself (’loops’) , we obtain a (pseudograph).

Figure 2.19: A pseudograph.

4. Allowing our edges to be arbitrary subsets of vertices (rather than just pairs) gives us hyper-graphs.

5. By allowing V (G) or E(G) to be an infinite set, we obtain infinite graphs.

If a graph has no loops or multiple edges, we call it a simple graph.

Exercise

1. Ten people are seated around a circular table. Each person shakes hands with everyone atthe table except the person sitting directly across the table. Draw a graph that models thesituation.

2.6 Degree of a Vertex

Definiton: The degree (or valence) of a vertex v in a graph G, denoted by deg(v), is the numberof proper edges incident on v plus twice the number of self-loops. ( Applications of graph theoryto physical chemistry motivate the use of the term valence).

Figure 2.20: A hypergraph.

Terminology: A vertex of degree d is also called a d− valent vertex.

Notation: The smallest and largest degrees in a graph G are denoted δmin and δmax (or δmin(G)and δmax(G) when there is more than one graph under discussion). Some authors use δ instead ofδmin and ∆ instead of δmax.

Definition. The degree sequence of a graph is the sequence formed by arranging the ver-tex degrees in non-increasing order.

Proposition 1. A non-trivial simple graph G must have at least one pair of vertices whose degreesare equal.

The work of Leonhard Euler (1707-1783) is regarded as the beginning of graph theory as a math-ematical discipline. The following result of Euler establishes a fundamental relationship betweenthe vertices and edges of a graph.

Theorem 4. Euler’s Degree-Sum Theorem.The sum of the degrees of the vertices of the graph is twice the number of edges.

Corollary 1. In a graph, there is an even number of vertices having odd degree.

Corollary 2. The degree sequence of a graph is a finite,non-increasing sequence of nonnegativeintegers whose sum is even.

Conversely, any non-increasing, nonnegative sequence of integers whose sum is even is the degreesequence of some graph. The following theorem prescribes how to construct such a graph. Thefollowing preliminary example illustrates the construction.

Theorem 5. Suppose that 〈d1, d2, · · · , dn〉 is a sequence of nonnegative integers whose sum is even.Then there exists a graph with vertices v1, v2, · · · , vn such that deg(vi) = di, for i = 1, 2 · · · , n.

Graphic Sequences

The construction in the above theorem is straightforward but hinges on allowing the graphto be non-simple. A more interesting problem is determining when a sequence is the degree se-quence of a simple graph.

Definition. A sequence 〈d1, d2, · · · , dn〉 is said to be graphic if there is a permutation of it thatis a degree sequence of some simple graph. Such a simple graph is said to realize the sequence.

Theorem 6. Let 〈d1, d2, · · · , dn〉 be a graphic sequence, with d1 ≥ d2 ≥ · · · ≥ dn. Then there is asimple graph with vertex-set {v1, · · · , v} satisfying deg(vi) = di for i = 1, 2, · · · , n, such that v1 isadjacent to vertices v2, · · · , vd1+1.

Corollary 3. [Havel (1955)and Hakimi (1961)]

A sequence 〈d1, d2, · · · , dn〉 of nonnegative integers such that d1 ≥ d2 ≥ · · · ≥ dn is graphic ifand only if the sequence 〈d2 − 1, · · · , dd1+1 − 1, dd1+2, · · · , dn〉 is graphic.

EXERCISES

1. Draw a graph with the given degree sequence.(a) 〈8, 7, 3〉 (b) 〈9, 8, 8, 6, 5, 3, 1〉.

2. Draw a simple graph with the given degree sequence.(a) 〈6, 4, 4, 3, 3, 2, 1, 1〉 (b) 〈5, 5, 5, 3, 3, 3, 3, 3〉.

3. Given a group of nine people, is it possible for each person to shake hands with exactly threeother people?

If d(x) = 0, then x is called an isolated vertex while a vertex of degree 1 is called a pendant.The edge incident with a pendant is called a pendant edge. A graph is called regular if all itsvertices have the same degree. If the common degree is r, it is called r-regular. In particular, a3-regular graph is called a cubic. We write δ(G) for the smallest of all degrees of vertices of G,and ∆(G) for the largest.

Theorem 7. In any graph or multigraph G , the sum of the degrees of the vertices equals twice thenumber of edges.

∑u∈V (G)

d(u) = 2 |E(G)| .

The above theorem is also known as ’Hand-Shaking Theorem’, from the following party analogy:Consider a collection of guests at a party. Suppose some guests shook hands with some otherguests. If we asked everyone at the party how many guests they shook hands with and added thosenumbers all up, this sum would be equal to twice the number of total hand shakes. In graph theoryterms, each vertex represents a guest, and an edge between two guests represents a handshakebetween them.

Corollary 4. In any graph or multigraph, the number of vertices of odd degree is even. In partic-ular, a regular graph of odd degree has an even number of vertices.

By the Hand-Shaking theorem, we obtain the following result.

Corollary 5. Every k − regular graph on n vertices has kn/2 edges. In particular, the completegraph Kn has (n− 1)n/2 edges.

Exercises:

1. Draw a portion of your academic department’s Web page as a graph by modeling Web pagesas vertices and hyperlinks as edges. Include at least ten different Web pages and draw edgesfor all of the existing hyperlinks between the Web pages you select. Label the vertices withtheir uniform resource locators.

2. Refer to the job assignments dicussed earlier. Is the company able to meet its hiring needs?If so, providr a possible set of hires that meet their needs.

3. For n ∈ {1, 2, 3, 4, 5}, draw the following graphs : Nn, Pn, Cn,Kn, and Kn,n. Which of thegraphs are the same?

4. Construct a table whose entries are formulas for the number of vertices and edges for thegraphs Nn, Pn, Cn,Kn and Km,n for arbitrary values of n and m.

5. Express both the maximum and minimum degree of vertices in the graphs Nn, Pn, Cn,Kn

and Km,n in terms of n and m.

6. For what kind of vertices u in a graph G do we have NG(u) = NG[u]?

7. For every k− regular graph, is there a (k + 1)- regular graph that contains the k− regulargraph as a subgraph?

8. Let m30(n) be the maximum number of edges that a simple graph on n vertices can have,if each vertex of the graph is connected to at most 30% of the other vertices. Determine aquadratic polynomial p(x) = Ax2 + Bx + C such that

(a) m30(n) ≤ p(n) for every n ∈ N .(b) m30(n) = p(n) holds for infinitely many n ∈ N .

Is p(n) uniquely determined by the two conditions above?

9. Let G be the simple graph, where the vertices correspond to each of the squares of an 8× 8chess board, and where two squares are adjacent if,and only if , a knight can go from onesquare to the other in one move. What is/are teh possible degree(s) of a vertex in G? Howmany vertices have each degree? How many edges does G have?

2.7 Connectivity

Given any graph G, the set of all edges of KV (G) that are not edges of G will form a graph withV (G) as vertex set; this new graph is called the complement of G, and written G. More generally,if G is a subgraph of H, then the graph formed by deleting all edges of G from H is called thecomplement of G in H denoted by H−G ( or the complement of G relative to H). The complementof the complete graph Ks on vertex set S is called a null graph; we also write Kv for a null graphwith v vertices.

A graph is called disconnected if its vertex set can be partitioned into two subsets, V1 and

V2, that have no common element, in such a way that there is no edge with one endpoint in V1 andthe other in V2; if a graph is not disconnected then it is connected.

A disconnected graph consists of a number of disjoint subgraphs; a maximal connected subgraph iscalled a component. In a way, connectedness generalizes adjacency. In a connected graph, not allvertices are adjacent, but if x and y are not adjacent, then there must exist vertices x1, x2, · · · , xn

such that x is adjacent to x1, x1 is adjacent to x2, · · · and xn is adjacent to y; such a sequence iscalled an xy− walk. Conversely, if every pair of nonadjacent vertices is joined by such a walk, thegraph is connected.

Theorem 8. For a simple graph G with n vertices and k components we have

|E(G)| ≤ (n− k)(n− k + 1)2

.

Chapter 3

Distance in Graphs

How far is it from one vertex to another? In this section we define distance in graphs, and weconsider several properties, interpretations, and applications.

Distance functions, called metrics, are used in many different areas of mathematics, and theyhave three defining properties.

If d is a metric, then

• d(x, y) ≥ 0, for all x, y, and d(x, y) = 0 if and only if x = y;

• d(x, y) = d(y, x) for all x, y;

• d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z.

The distance concept in the graph sense is in fact a metric.

Distance in graphs is defined in a natural way: in a connected graph G, the distance from vertexu to vertex v is the length (number of edges) of the shortest u − v path in G. We denote thisdistance by d(u, v), and in situations where clarity of context is important, we may write dG(u, v).

In Figure 4.1 , d(b, k) = 4 and d(c,m) = 6.

Figure 3.1: A graph G

For a given vertex v of a connected graph, the eccentricity of v, denoted ecc(v), is defined to bethe greatest distance from v to any other vertex. That is

ecc(v) = maxx∈V (G) {d(v, x)}

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In Figure 4.1 ecc(a) = 5 since the farthest vertices from a (namely k,m,n) are at a distance of 5from a.

Of the vertices in this graph, vertices c, k, m and n have the greatest eccentricity (6), and ver-tices e, f and g have the smallest eccentricity (3). These values and types of vertices are givenspecial names.

In a connected graph G, the radius of G, denoted rad(G), is the value of the smallest eccen-tricity. Similarly, the diameter of G, denoted diam(G), is the value of the greatest eccentricity.

The center of the graph G is the set of vertices v, such that ecc(v) = rad(G). The peripheryof G is the set of vertices u, such that ecc(u) = diam(G). In Figure, the radius is 3, the diameteris 6, and the center and periphery of the graph are, respectively, {e, f, g} and {c, k, m, n}.

You may have noticed that the diameter of our graph G is twice the radius of G. While thisdoes seem to be a natural relationship, such is not the case for all graphs. Take a quick look at acycle or a complete graph. For either of these graphs, the radius and diameter are equal!

The following theorem describes the proper relationship between the radii and diameters of graphs.While not as natural, tight or ’circle-like’ as you might hope, this relationship does not have theadvantage of being correct.

Theorem 9. For any connected graph G, rad(G) ≤ diam(G) ≤ 2rad(G).

Theorem 10. Every graph is (isomorphic to) the center of some graph.

Exercises.

1. Find the radius, diameter and center of the graph shown below.

2. Find the radius and diameter of each of the following graphs: P2k, P2k+1, C2k, C2k+1,Kn,Km,n.

3. If x is in the periphery of G and d(x, y) = ecc(x), the prove that y is in the periphery of G.

4. Prove that if G is regular and diam(G) = 3, then diam(G) = 2.

Chapter 4

Trees

A tree is a connected graph that contains no cycles. Graph-theoretic trees resemble the trees wesee outside our windows. Trees do not have cycles, just as the branches of trees in nature do notsplit and rejoin.

Figure 4.1: Which one are trees?

The graph D in Figure is not a tree; rather, it is a forest. A forest is a collection of one or moretrees. A vertex of degree 1 in a tree is called a leaf. K1 and K2 are considered as trees. In thespirit of our terminology, perhaps we should call K1 a stump and K2 a twig !

It is clear that K1 and K2 are the only trees of order 1 and 2, respectively. P3 is the only tree oforder 3.

The Figure below shows the different trees of order 6 or less.

Trees sprout up as effective models in a wide variety of applications.

Example

1. Trees are useful for modeling the possible outcomes of an experiment. Consider an experimentin which a coin is flipped and a 6-sided die is rolled. Tne leaves in the tree in Figure correspond

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Figure 4.2: Trees of order 6 or less.

to the outcomes in the probability space for this experiment.

Figure 4.3: Outcomes of a coin/die experiment

2. Programmers often use tree structures to facilitate searches and sorts and to model logic ofalgorithms. For instance, the logic for a program that finds the maximum of four number5s(w,x,y,z) can be represented by the tree shown in Figure. This type of tree is a binarydecision search tree.

3. Chemists can use trees to represent, among other things, saturated hydrocarbons- chemicalcompounds of the form CnH2n+2 (propane, for exmple). The bonds between between thecarbon and hydrogen atoms ate depicted in the trees of Figure. The vertices of degree 4 arecarbon atoms, and the leaves represent the hydrogen atoms.

Figure 4.4: Logic of a program

Figure 4.5: A few saturated hydrocarbons.

Exercises

1. Draw all unlabeled trees of order 7. (Hint: There are a prime number of them.)

2. Draw all unlabeled forests of order 6.

3. Let T be a tree of order n ≥ 2. Prove that T is bipartite.

4.1 The Minimum Spanning Tree Problem

Suppose that initially, no roads existed between any pair of the villages v1, v2, · · · , v7. Then weneed to construct roads between pairs of dormitories. Which roads will be constructed is quitepossibly a financial decision here as well. Before proceeding further, let us consider a new concept.

If a connected graph G of order n has no cycles, then of course, G is a tree. On the other hand,suppose that G contains cycles. Recall that a subgraph H of a graph G is a spanning subgraphof G if H contains every vertex of G. A spanning subgraph H of a connected graph G such that His a tree is called a spanning tree of G. For the connected graph G, two different spanning treesT1 and T2 of G are shown.

Figure 4.6: Two spanning trees in a graph

Figure 4.7: Villages and roads

Once again, let us return to the example we considered in Figure 4.7, where there are seven villagesand six roads. The graph describing this situation is a tree. Hence it is possible to travel betweenevery two villages. Indeed, there is a unique path between every two villages. To travel betweenvillages v1 and v6, we are forced to pass through v2, v3, and v4, even if we didn’t want to. Therefore,the trip between v1 and v6 may be inconvenient. Of course, to make the trip between pairs of thevillages more convenient , we could always build a new road (between v1 and v6, say). However,this would cost more money (possibly a great deal of money). But how was it decided initially thatthe six roads in Figure 5.7 were the ones to be constructed? Certainly whichever roads were chosenshould produce a connected graph. If the resulting graph contains a cycle, then there are edges inthe graph that are not bridges. That is, if producing a connected graph was our primary goal, thewe could have accomplished this for less money by constructing roads so that the resulting graphis a tree. But how did we chose those particular roads to build?

Figure 4.8: A graph model of villages and roads.

Suppose that we have a number of villages (such as v1, v2, · · · , v7) and we would like to buildroads as cheaply as possible so that, at the conclusion, the resulting graph is connected. How dowe do this?

Assume that we have an accurate estimate of the cost of building a road between each pair ofvillages. If the cost of building a road between some pair of villages is exorbitant ( because anysuch road would have to pass through quicksand, private property, or through or over a mountain,for example), then we do not even consider building such a road. This problem can be stated interms of graphs.

Let G be a connected graph each of whose edges is assigned a number (called the cost orweigth of the edge). We denote the weight of an edge e of G by w(e). Recall that such a graphis called a weighted graph. For each subgraph H of G, the weight w(H) of H is defined as thesum of the weights of its edges, that is

w(H) =∑

e∈E(H)

w(e).

We seek a spanning tree G whose weight is minimum among all spanning trees of G. Such a span-ning tree is called a minimum spanning tree. The problem of finding a minimum spanning treein a connected weighted graph is called the Minimum Spanning Tree Problem.

The importance of the Minimum Spanning Tree Problem is due to its applications in the designof the computer, communications, and transportation networks.

Over the years, this problem has been solved in a variety of ways using a number of algorithms.One of the best known was discovered by John Bernard Kruskal.

Kruskal’s Algorithm

For a connected weighted graph G, a spanning tree T of G is constructed as follows: Forthe first edge e1 of T , we select any edge of G of minimum weight and for the second edge e2 of T ,we select any remaining edge of G of minimum weight. For the third edge e3 of T , we choose any

remaining edge of G of minimum weight that does not produce a cycle with the previously selectededges. We continue in this manner until a spanning tree is produced.

Illustration

Chapter 5

Eulerian Trails and Circuits

Recall that a trail in a graph is a walk that does not repeat any edges, and that a closed trail (onethat begins and ends at the same vertex) is called a cicrcuit.

If a trail in a graph G includes every edge of G, then that trail is said to be an Eulerian trail.Similarly, an Eulerian circuit in a graph is a circuit that includes every edge of the graph. Agraph that contains an Eulerian circuit is said to be an Eulerian graph.

There are two-well-known characterizations of Eulerian graphs. One involves vertex degrees, andthe other concerns the existence of a special collection of cycles.

Theorem 11. For a connected graph G, the following statements are equivalent.

1. G is Eulerian.

2. Every vertex of G has even degree.

3. The edges. of G can be partitioned into (edge-disjoint) cycles

Corollary 6. The connected graph G contains an eulerian trail if and only if there are at mosttwo vertices of odd degree.

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Chapter 6

Modeling Using Graph Theory

6.1 Graph as Models

The Seven Bridges of KonigsbergEuler’s Problem (restated)

Given a graph, under what conditions is it possible to find a closed walk that traverses every edgeexactly once?

We will call graphs for which this kind of walk is possible Eulerian.

What graphs are Eulerian? If you think for a moment, it is pretty clear that the graph has tobe connected - that is, there must be a path between every pair of vertices. Another observationyou might make is that whenever you walk over bridges between land masses and return to thestarting point, the number of times you enter a a land mass is the same as the number of timesyou leave it. If you add the number of times you enter a specific land mass to the number of timesyou leave it. If you add the number of times you enter a specific land mass to the number of timesyou leave it, you therefore get an even number of eges incident with each vertex is needed. In thelanguage of graph theory, we say that every vertex has even degree or that the graph has even degree.

Thus we have reasoned that Eulerian graphs must be connected and must have even degree. Inother words, for a graph to be Eulerian, it is necessary that it both be connected and have an evendegree. But it is also true that for a graph to be Eulerian, it is sufficient that it be connected witheven degree. Establishing necessary and sufficient conditions between two concepts- in this case,”Eulrian graphs” and ”all connected graphs with even degree ” - is an important idea in mathe-matics with practical consequences. Once we establish that being connected with even degree isnecessary and sufficient for a graph to be Eulerian, we need only model a situation with a graph,and then check to see whether the graph is connected and each vertex of the graph has even degree.

Graph Coloring

Four-Color Problem Given a geographic map, is it possible to color it with four colors so thatany two regions that share a common border (of length greater than 0) are assigned different colors?

39

Figure 6.1: United States map

The four-color problem can be modeled with a graph. Figure 6.2 shows a map with a vertex rep-resenting each state in the continental United States and an edge between every pair of verticescorresponding to states that share a common (land) border. Note that Utah and New Mexico, forexample, are not considered sdjacent because their common border is only a point.

Figure 6.2: United States map with graph superimposed

The figure below shows the graph only. Now the original question we posed about the map ofthe United States has been transformed into a question about a graph.

Four-Color Problem (restated) Using only four colors, can you color the vertices of a graph sothat no vertex gets the same color as an adjacent vertex?

We’ll call a coloring proper if no two adjacent vertices share the same color.

Figure 6.3 shows a solution to the four-color problem on the graph in Figure 6.2.

Figure 6.3: Graph-coloring solution

Application of Graph Coloring

One problem typically modeled with graph coloring is final exam scheduling. Suppose that auniversity has n courses in which a final exam will be given and that it desires to minimize thenumber of exam periods used, while avoiding ’conflicts’. A conflict happens when a student isscheduled for two exams at the same time. We can model this problem using a graph as follows.We start by creating a vertex for each course. Then we draw an edge between two vertices when-ever there is a student enrolled in both courses corresponding to those vertices. Now we solve thegraph-coloring problem; that is, we properly color the vertices of the resulting graph in a way that

minimizes the number of colors used. The color classes are the time periods. If some vertices arecolored blue in a proper coloring, then there can be no edges between any pair of them. This meansthat no student is enrolled in more than one of the classes corresponding to these vertices. Theneach color class can be given in its own time slot.

Problems

1. Solve the softball manager’s problem (both versions) from the Introduction to this chapter.

2. The bridges and land masses of a certain city can be modeled with the graph G shown below.

(a) Is G Eulerian? Why or why not?(b) Suppose we relax the requirement of the walk so that the walker need not start

and end at the same land mass but still must traverse every bridge exactly once. Is thistype of walk possible in a city modeled by the graph? If so, how? If not, why not?

3. Consider the graph shown below.

(a) Color the graph with three colors.(b) Now suppose that vertices 1 and 6 must be colored red. Can you still color the graphwith three colors (including red) ?

4. The Mathematics Department at a small college plans to schedule final exams. The classrosters for all the upper-class math courses are listed in the table below. Find an examschedule that minimizes the number of time periods used.

Course StudentsMath 350 Jimi B.B. EricMath 365 Ry Jimmy P. CarlosMath 385 Jimi Chrissie Bonnie BrianMath 420 Bonnie Robin CarlosMath 430 Ry B.B. Buddy RobinMatn 445 Brian BuddyMath 460 Jimi Ry Brian Mark

Miscellaneous Problems

1. Consider the graph in the figure.(a) Write down the set of edges of E(G).(b) Which edges are incident with vertex b?(c) Which vertices are adjacent to vertex c?(d) Compute deg(a).(e) Compute |E(G)|.

2. At a large meeting of business executives, lots of people shake hands. Everyone at the meetingis asked to keep track of the number of times she or he shook hands, and as the meeting ends,these data are collected. Explain why you will obtain an even number if you add up all theindividual handshake numbers collected. What does this have to do with graphs? Expressthis idea using the notation introduced in this section.

Chapter 7

Selected Topics

7.1 Domination in Graphs

TERMINOLOGY. A vertex is said to dominate itself and each of its neighbors.

The applicability of domination theory to such fields as network design and analysis, linearalgebra, and optimization has attracted researchers from a broad range of disciplines. The originsare ascribed to C. Berge and O. Ore.

Definition. Let G be a graph and let D ⊂ V (G). A vertex subset D dominates a graph G(or is a dominating set) if every vertex of G if every vertex of G is in D or is adjacent to at leastone vertex in D.

Definition. A minimal dominating set of a graph G is a dominating set such that everyproper subset is non-dominating.

Definition. The domination number of a graph G, denoted dom(G) ( elsewhere, often Γ(G)))),is the cardinality of a minimum dominating set of G.

Example. Figure shows a graph with minimal dominating sets (the solid vertices) of two dif-ferent cardinalities. It is straightforward to show that the dominating set on the left is a minimumone (i.e, there are no 2-vertex dominating sets). Thus, the dominating number of the graph is 3.

Figure 7.1: Two minimal dominating sets for a graph.

45

Example. For the 5-vertex path graph P5, we have dom(P5) = 2. The second and fourth verticeson the path form the unique minimum dominating set.

Application. Chess.

Consider the problem of placing queens on a chessboard so that every square is either occupied bya queen or can be reached in one move by a queen. Determining the minimum number of queens isequivalent to finding the domination number of a 64− vertex graph, where two vertices are adjacentif and only their corresponding squares lie on the same diagonal, same row, or same column.

Remark: Both dominating set in Figure are minimial, because each vertex in each set domi-nates at least one vertex that no other vertex in that set dominates.

Exercises.

1. Find all minimum dominating sets of the following graph, and argue why there are no smallerones.

2. Determine the domination number of (a) the n-vertex path graph Pn, (b) The n-vertex cyclegraph Cn.Answer: dom(Pn) =

⌈n

3

⌉, since each vertex can dominate at most three vertices.

3. Determine the minimum number of knights that can be placed on a chessboard so that eachsquare is either occupied by a knight or can be reached in one move by a knight.

References

1. Harris, J., Hirst, J.,Mossinghoff, M.,Combinatorics and Graph Theory, Second Edition, Springer,2008.

2. Agnarsson, G. and Greenlaw, R., Graph Theory: Modeling, Applications and Algorithms,Pearson Education,Inc., 2007.

3. Wallis, W.D., A Beginner’s Guide to Graph Theory, 2nd Ed., Birkhauser Boston, 2007.

4. Gross, J. and Yellen, J., Graph Theory and Its Applications 2nd Ed., Chapman and Hall/CRC,2006.

5. Chartrand, G. and Zhang, P. Introduction to Graph Theory, McGraw Hill, InternationalEdition, 2005.