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METU DEPARTMENT OF MATHEMATICS

GRAPH THEORY

LECTURE NOTES

Ali Doanaksoy

Ankara, 1997

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I- PRELIMINARIES1- Definition. A graph is a pair consisting of a finite(1)

nonempty(2) set

and a finite(3) collection

of distinct(4)

unordered(5) pairs of distinct(6) elements of Elements of arecalled verticesand elements of are called edgesof G. Thus, we call the vertex set and , the edge set. An edge is denoted by where and v are vertices. If the graph G is clear for the given context, we writebriefly and to denote and respectively. The number ofelements of is called the order and the number of elements of iscalled the sizeof the graph. We denote the order of the graph by

(or

or

); and size by (or or ).2- Plane representation. A representation of a graph in plane is a figureconsisting of points and line segments. We draw this figure such that

- to each point in figure there corresponds a unique vertex of andconversely,

- a line segment is drawn between two points corresponding the vertices and

if and only if

is in the edge set ofG.

An edge is said to join the vertices and . It is also said that isincident to and .Ifuand v are vertices of a graph and is an edge, we say that and areadjacent, and that is a neighbourof, is a neighbour of. If and areadjacent vertices, we write . The set of all vertices which are adjacent toa vertex is called the neighbourhood ofTwo edges incident to a same vertex are also said to beadjacent.

Example. Let be the graph with the vertex set and theedge set Then . A visualrepresentation is as follows:

a

b c

d

ab ac

cd

bc

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3- Variations on the definition. We may have variations of the definition of agraph by replacing the numbered statements in (1). If (1) is replaced withinfinite we obtain graphs with infinite vertices So dropping (1), we let agraph to have finitely or infinitely many vertices. (2) can never be dropped.

That is, the order of any graph is at least 1. (3) can be omitted to let graphs topossess infinitely many edges. If(4) is not required, we can have multiple (or

parallel) edges joining the same pair of vertices. If (5) is replaced withordered the resulting structure is a directed graph When (6) is dropped,edges joining a vertex to itself may appear. Such an edge is called aloop.

We consider only finite graphs, namely the graphs with || and||. So throughout the text, (1), (2) and (3) in definition (1) will neverbe changed or dropped. In some cases we may find it useful to replace (5)

with ordered In such a case we call the graph adirected graphor a digraph.

In many cases it may be necessary to consider the graphs with (4) and (6) aredropped in (1). If (4) and/or (6) are not required, we call the graph a

multigraphor apseudeograph.

(5): unordered(4): distinct(6): distinct

(5): ordered(4): distinct(6): distinct

(5): unordered(4): -(6): distinct

(5): ordered(4): -(6): distinct

(5): unordered(4): -(6): -

(5): ordered(4): -(6): -

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5- Independent set of vertices. A set of vertices no two are adjacent is called anindependent set of vertices. The cardinality of a largest independent set of

vertices is called the independence numberof the graph.

Example. Let the vertices of the graph represent 6 students and let an edgeof to stand for the friendship relation. An independent set of vertices ofthis graph is a subset of the students that no one knows the others. For

example is not an independent set but are. The largest ofsuch subsets contain 3 members, so has independence number 3.

a

b

c

d

e

f

Example. Now suppose that the six students of previous example arematched two by two. Naturally none of them would like to be matched with

another who is not known by her. Say, if we have the matchings and then, and are to stay alone. If we wish to have a complete matching,we may consider and . As it is seen, to have a matching ofvertices is equivalent to have a set of edges no two are adjacent. If we

consider the below labeling of the vertices, the first matching coreesponds to

the set and the latter one corresponds to .a

b

c

d

e

f

e1e

2

e3e 4

e5

e6

6- Matching. A set of edges no two are adjacent is called an independent set ofedges or a matching. The size of the largest matching is called the edge-

independence number of . An independent set of edges covering all thevertices is called a 1-factor or a complete matchingof

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a 1-factor in Gno complete matchings possibleedge-independence number: 3

7- Labeled graph. A labeled graphof order is a graph to each vertex of whichan integer is assigned such that no two vertices are assigned thesame number. A graph with a distinguished vertex is called a rooted graph.

8- Local degree (Valency). For each vertex in a graph , the number of edgesincident to is called the valency of or the local degree of at If thegraph in question is a multigraph, then a loop at vertex contributes 2 to thevalency of. We denote the valency of by or . By and , the maximum and minimum valencies in are denotedrespectively. A vertex with valency 0 is called an isolated vertexand a vertex

with valency 1 is called an end-vertex (or a leaf ). The (non-decreasing)valency sequence of is the set of valencies (arranged in non-decreasingorder).

isolated vertex

end vertex

valency sequence: 0,1,1,2,3,3,4

9- If we add all valencies in a graph , the resulting number gives twice thenumber of edges of . Because, each edge has two end points which are

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EXERCISES

I.1 For any even integer show that there exists a regular simplegraph.

I.2 Let and be positive integers such that is even.Showthat there existsa regular graph of order I.3 Show that a sequence of non-negative integers is valencysequence of a (not necessarily simple) graph if and only if their sum is even.

I.4 Show that there exists no simple graph whose valency sequence is2,3,3,4,5,6,7 or 1,3,3,4,5,6,6,.I.5 If is valency sequence of some simple graph with then, the sum is even and

i n

for [In fact, in 96 Erds and Gallai has proved that this necessary condition is also sufficient] I.6 For any graph show that I.7 Prove that any simple graph has at least two vertices with the same valency.I.8 Show that a simple graph with for some positive integer ,contains a matching of size

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II GRAPH ISOMORPHISM12-Isomorphism. Two graphs and are said to be isomorphic (written

if there exists a 1-1 correspondence

such that for

any if and only if .Example. Given :

and

.

a

b

c

d

e

f

p

q

r

s

t

u

Then the mapping given by

a

b

c

d

e

f

p

pq

q

r

r

s

s

tt

u

u

is an isomorphism.

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Let and be graphs. If for an arbitrary labeling of it is possible to find alabeling of in such a manner that , then and areisomorphic. For the graph in above example we have

a

b

c

d

e

f

p

q

r

s

t

u

1

1

2

2

33

4

4

5

5

6

6

so that 613- Automorphism group of a graph. An isomporphism of a graph onto itself is

called an automorphismofExample.

a a

a

b b

bc c

c

d d

d

e e

ef f

fg g

gh h

h

Compositions of automorphisms are again autmorphisms. In fact, the set of automorphisms of is a group, which is called the automorphismgroupof. is transitive if it contains transformations which map eachvertex of to every other vertex.

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14-Some necessary (but not sufficient) conditions for isomorphism of twographs are as follows.

Let and be two isomorphic graphs. Then,a-

|| ||,b- || || ,

c- Non-decreasing valency sequences of and are same.The following example figures out that the converse statements are not true:

Examples. For the following graphs and , || | | ,|| | | and valency sequence of both is but

G H

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EXERCISESII.1 Determine all simple graphs of order 4 up to isomorphism.II.2 Determine all simple regular graphs of order 6 up to isomorphism.II.3 Let and be the plane graphs given below. Show that

II.4 Let and be the plane graphs in the figure.Show that II.5 For the graphs and in the figure, compute and II.6 Determine all simple graphs with valency sequence 1,2,2,3,3,3 up toisomorphism.

II.7 Call two graphs edge isomorphic if there exists a bijection between theedge sets which preserves the adjacency of edges. Show that edge isomorphism

does not imply isomorphism.

II.8 A graph is said to be vertex transitive (or symmetric) if for any pair ofvertices, there is an automorphism of the graph which maps one of these vertices

to the other. Show that all simple graphs on five or less vertices are symmetric.

Give an example of an asymmetric simple graph on six vertices.

II.9 A graph is said to be edge transitiveif for any pair of edges, there is an edgeautomorphism of the graph which maps one of these edges to the other. Find

a) a graph which is edge transitive but not vertex transitive,b) a graph which is vertex transitive but not edge transitive.

G1 G2 G3

G H

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III SPECIAL GRAPHS

15-Complete graph. A simple graph in which every pair of vertices are joined byan edge is called a complete graph. A complete graph of order is a valent graph and has edges. We denote this graph by .Since among all simple graphs of order , is the one which have thelargest possible number of edges, there exists no simple graph with . is usually represented in plane by an gon with all itssides and diagonals being drawn.

K1 K2 K3 K4 K5 K6

16-Circuit graph. The graph whose vertices are vertices of an gon and whoseedges are the sides of the gon is called the circuit graph of order. In otherwords

is the unique connected graph of order

which is

valent.

Thus for,

C1 C2 C3 C4 C5 C6

17-Path graph. The path graph on vertices is defined for , the graphwhich is obtained by deleting an edge of the circuit graph and .

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P4

P2

P3

P5

P6

18-Wheel graph. By adding a new vertex to the circuit graph which is joinedto all other vertices, the wheel graph is obtained.

W2

W3

W4

W5

W6

19-Bipartite graph. A bipartite graph is a graph whose vertex set is partitionedinto two parts such that no two vertices in the same part are adjacent.

The complete bipartite

is a bipartite graph with vertex set partitioned

int two sets of sizes and , each vertex in one part is joined to all vertices inthe other part. Some examples are as follows:

K1,1

K1,2

K2,2 K2,3

K3,3

has

vertices and

edges. The graph

is called a star graph.

K1,8

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20-A complete partite graph is obtained by partitioning the vertex set into sets and joining two vertices by an edge if and only if the are in different

parts. If not all such vertices are joined, we obtain an partite graph.

A 3- partite graph A complete3-partite graph

If all the sets of vertices in a complete partite graph have the same size , thegraph is denoted by .

K3(2)

Note that has vertices an it is an valent graph, hence it has edges.21-Polyhedral graphs. Any graph drawn on the sphere can be mapped to a

(unique) graph in plane via stereographic projection and conversely. In

particular each polyhedron corresponds to a graph. In particular the five

regular polyhedral (platonic solids) can be thought as graphs with vertices

being on a sphere.

Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron

Graphs which correspond to regular polyhedral are called the regular

polyhedral graphs(orplatonic graphs)

Tetrahedr on Hexahedron Octahedron Dodecahedron Icosahedron

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22- dimesional octahedron. It is seen that the octahedron is the graph .

4 dimensional octahedron: (2)K4 Octahedron: (2)K3

The graph is defined to be the dimensional octahedron.

4 dimensional octahedron: (2)K4

23- dimensional cube. Note that the vertices of the hexahedron (cube) graphcan be labelled with the binary triples 000,001,010,011,100,101,110,111 in

such a manner that labels of two adjacent vertices differ only in one

coordinate.

000 001

010 011

100 101

110 111

As a generalization, we define the dimensional cube(or cube) to bethe graph with vertices which are labeled with binary tuples and twovertices are adjacent if their labels differ only in one coordinate.

4-cube

EXERCISESIII.1 Show that cube is bipartite.III.2 Show that if is a simple graph with where and arerespectively the number of edges and vertices of then cannot be bipartite.

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IV OBTAINING NEW GRAPHS FROM A GIVEN GRAPH.24-Complement of a graph. Given a simple graph complement of is the

graph with

and two vertices of which are adjacent if they are

not adjacent in . In other words, is such a graph that , and where is the order ofExample.

G G

25-A graph which is isomorphic to its complement is called a self-complementarygraph.

Example.

G G

Proposition. Order of a self-complementary graph is of the form or for some positive integerProof. Let be a self-complementary graph of order Since wehave | | || . But so| | || | | and since we get || ||which gives || , that is

||

and

cannot both be even so, for

||to be an integer, either

or

must be a multiple of 4. Corollary. The number of edges of a self-complementary graph of order graph of

order is

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Repeating this for all the edges of we obtain the following graph.

( )L K5

This graph is known as Petersen graph and can be re-drawn as follows:

Pet ersen G raph

28-Total graph. The total graph of a graph is the graph whose verticescorrespond to the edges and vertices of and two vertices of which are joine ifand only if the corresponding vertices or edges of are adjacent or incident.

G L G( ) T G( )

29-Subgraph. If

and

are graphs with

and

we say

that is a subgraph of , denoted A subgraph of with is called aspanning subgraph ofExample.

Graph

Subgraphs

Spanning subraphs

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A subgraph ofG such that two vertices ofHare a adjacent whenever they areadjacent in is called an induced subgraph.Example.

Graph Induced subgraph

If is an induced graph of with the vertex set , we say that is a subgraphof induced by Example. Consider the graph in the previous example.

A subsetof the vertex set

W Subgraph induced by W

30-Clique, triangle. A cliquein a graph is a complete subgraph of.

Clique K4

A clique is also called a triangle.31-Edge and vertex deletion. Given a graph . By removing an edge of we

obtain the edge-deletedsubgraph

. For a vertex

of

, the vertex deleted

subgraph is obtained from by deleting the vertex together with all edgesincident to

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Example.

G G- e G-v

e

v

32-Although the definition does not allow a graph to have an empty vertex set,after some successive operations all the vertices of might be removed. Inthat case we say that the graph is empty.

33-Contraction.Removing an edge of a graph and identifying the vertices and is called contracting the edgee. The resulting graph is denoted by Ifthe graph can be obtained from by contracting some edges, is said to becontractible to

Example.

G G\e

e

In general it is not true that for two distinct edges of If and of a graph 34-Homeomorphic graphs. Replacing an edge

by two new edges

and

, where is a new vertex is called inserting a vertexinto an edge. If twographs can be obtained from the same graph by inserting vertices into itsedges, they are called homeomorphic.

Example.

Homeomorphic graphs

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EXERCISESIV.1 Show that for any simple graph .IV.2 Show that line graph of the following graph is isomorphic with thecomplement of the graph.

IV.3 Show that can never be an induced subgraph of a line graph.IV.4 Show that

.

[In fact, other than and , for any graphs and , isomorphism of linegraphs implies isomorphism of the graphs.]IV.5 Let be a simple graph with || and || . Let be thenumber of triangles in Show that

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38-Cartesian Product. The Cartesian product of and is defined bysetting and if and only if and ] or and ]Example.

G

H

a

b

c

d

x y z

(a,x)(a,y)

(a,z)

(b,y)

(b,x)

(b,z)(c,x)

(c,y)

(c,z)

(d,x)

(d,y)

(d,z)

G Hx

Example. For any graph and for any positive integer Cartesian product of two copies of the same graph is denoted by andand for any positive integer is defined to be the graph

39-Composition. The composition of and of and is the graph ]defined as ] and if and only if] or ].Example.

P2

P2[ ]P3

P3

P2

P3[ ]P2

P3

Note that, in general ]].

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40-Conjuction. The conjuctionof and of and is the graph definedas and if and only if] and ].

P2

P2 P3

P3V

EXERCISESV.1 Give examples of graphs and to demonstrate thata) and regular does not imply that is regular,

b) and bipartite does not imply that is bipartite,V.2 For any two graphs and , show that it is never true that

a) b) c)

] ].

V.3 a) Show that the number of edges of the cartesian product of two graphsmay be a prime number.

b) Show that the number of edges of the conjuction of two graphs is

necessarily an even number.

V.4 Let|| , || , || and || . Show thata) | | ,b) | | c) | | d) |]|

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VI. WALKS, TRAILS, PATHS AND CYCLES

41- Walk, trail, path. In a graph

a sequence of vertices

such that

and are adjacent for is called a walkoflength from to in A walk consisting of a single vertex is called a trivial walk. Awalk is called a trailif no edge is repeated and if no vertex is repeated it is

called apath. If the length of a walk is even (resp. odd), it is called an even

(resp. odd) walk.

a walk: abcfhgfcde a path: abcfhge

a

b

c

d

e

f

g

h

i

a trail: abcfhgfde

a

b

c

d

e

f

g

h

i

a

b

c

d

e

f

g

h

i

42-Closed walk, tour, cycle. Let be a walk in a graph . If then the walk is said to be closed, otherwise it is called open. A closed trail is

called a tour. Although in a path, repetition of vertices is not allowed, by

letting exceptionally, we obtain a closed pathwhich is also called acycle(or a circuit).

a cycle: abfdehga

a

b

c

d

e

f

g

h

a closed trail: abcdebfga

i

a

b

c

d

e

f

g

h

i

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43-THEOREM. A graph is bipartite if and only if it contains no odd cycles.Sketch of proof. Suppose that is bipartite with being the partition ofthe vertex set. Consider a cycle

. We may assume that

Since is bipartite, it follows that , . It is seen that is even and length of is even.Now assume that has no odd cycles. Pick an arbitrary vertex and put it ina set and put all vertices adjacent to in a set If two vertices in areadjacent, together with , these vertices constitute a triangle which is an oddcycle. So, no two vertices in Ycan be adjacent. Now, take all vertices in which are adjacent to some vertex in

and put then in

Continuing in this

manner we obtain a partition ) of the vertex set so that is bipartite.44-Distance, eccentricity, radius, diameter, girth. The distance between two

vertices of a graph is the length of the shortest path joining these two

vertices. If there is no such path, we write to mention this case. Thegreatest distance measured at a vertex is the eccentricity of thevertex

. Radius

of the graph is the minimum of all eccentricities of the

vertices. A vertex with is called a centralvertex. Maximum of alleccentricities is the diameter of the graph. In other words, diameter is themaximum distance which can be measured between two vertices of the

graph.

Example. : radius 1, diameter 2 (

radius 2, diameter 2 (

: radius 1, diameter 1 : radius] diameter ] ( : radius diameter ] (

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Example. Consider the graph given in the following figure. Except the pairs and distance between any pair of vertices is either 1 or 2. It followsthat and . Then diameter is 3 and radius is 2.

a

b

c

d

e

f

g

h

i

Diameter: 3Radius 2

45-Girth, circumference. Lengths of a shortest cycle and a longest cycle in arecalled respectively thegirthand the circumferenceof.Example. : no cycles

girth 4, circumference

in (

: girth 3, circumference : girth , circumference ( : girth 3, circumference ( Example.

Girth: 3 Circumference: 9

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EXERCISESVI.1 Show that if there is a walk from t o in , then there is also a path from t o in VI.2 Show that if is simple and , then has a path of length k.VI.3 Show that if is simple with order diameter 2 and then VI.4 Show that if an edge is in a closed trail of, then is in a cycle ofVI.5 Show that if is simple and [ , then is connected.VI.6 Show that if is connected and all vertices are even, then for any vertex the number of components of

is at most

VI.7 Show that any two longest paths in a connected graph have a vertex incommon.

VI.8 Prove triangle inequality for the distance function defined on graphs.That is, prove that for any three vertices inany graph VI.9 Show that if is simple with diameter 2 and then VI.10 Show that if diameter of a graph is greater than 3, then diameter of thecomplement is less than 3.

VI.11 Show that if an edge is in a closed trail of, then is in a cycle ofVI.12 Show that if is simple and , then contains a cycle of length atleast VI.13 Show that a regular graph of girth 5 has at least vertices.VI.14 Show that a regular graph of girth 5 and diameter 2 has exactly vertices and find such a graph for

and

[In fact Hoffman and Sinleton, 1960 have shown that such a graph can exist only if ]VI.15 Show that if , contains a cycle.VI.16 Show that if contains two edge-disjoint cycles.

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VII. CONNECTEDNESS46-Connected graph, component. In a graph , if there exists a path between any

pair of vertices,

is said to be connected. A graph which is not connected is

disconnected. In a disconnected graph, each connected maximal subgraph

(that is a subgraph which is not contained in any other connected subgraph)

is called a componentof. We denote the number of components by 47-THEOREM.If for a graph , then is disconnected.48-THEOREM. If , then is connected.49-Connectivity. A graph with

is said to be

connected if any pair

of vertices are connected by at least pairwise independent paths. Theconnectivity of, denoted by , is defined to be the largest value of forwhich is connected.A graph is edge connectedif every pair of vertices are connected by atleast edge disjoint paths. The largest value of for which is edgeconnected is denoted by and is called the edge-connectivity. Fromdefinitions it follows that

.

Example. : , (each edge is a bridge) i n ( : : ( : 2 ( 50-MENGERS THEOREM. Let

be a connected graph with

Then

a) is connected if and only if cannot be made disconnected by theremoval of or fewer vertices,

b) is edge connected if and only if cannot be made disconnected bythe removal of or fewer edges,

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51-Cutpoint, block. In a connected graph if removal of a vertex makes adisconnected graph, is called a cut point(or cut vertex). In general a vertex of a graph is called a cut pointof if the number of components of is larger than the number of components of

. If

is connected and

is disconnected, the set is called aseparating set of vertices.

cutpointseparating set of vertices

(connectivity: 2)

A connected non-trivial graph is called non-separable. In other words agraph is non-separable if it has no cut points. A blockof a graph is a maximal

non-separable subgraph It follows fro Mengers Theore thatconnectivity of is the minimum number of vertices whose removaldisconnects (or reduces to a single vertex).

52-Bridge. Similarly if for an edge the number of components of islarger than the number of components of is called a bridgeof A cutsetis a set of edges whose removal disconnects It follows fro MengersTheorem that, edge connectivity of is the minimum number of edgeswhose removal disconnects .

bridgecutset

(edge connectivity: 3)

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53-Strongly regular graph. A directed graph is called strongly regular if fromeach vertex there is a directed path to any other vertex.

54-THEOREM. If

is a spanning subgraph of

, then

.

55-THEOREM Whitneys Inequality. For any graph , .56-THEOREM. If ] then 57-THEOREM. If , then .58-THEOREM. For any graph with order

a)

b) where { if if

EXERCISESVII.1 Show that if a connected graph

has no separating sets, then it is a

complete graph.

VII.2 Show that each graph has a bipartite subgraph such that|| ||VII.3 Show that an edge is a bridge if and only if it is not in any cycle.VII.4 Show that if an edge is a bridge and then at least one of the

endpoints is a cutpoint.

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VIII. EULERIAN TOURS , HAMILTONIAN CYCLES59-Seven bridges of Knigsberg. The city of Knigsberg in Prussia

(nowKaliningrad, Russia) was set on both sides of the Pregel River together

with two large islands which were connected to each other and the other

parts of the city. The people wondered if it is possible to walk around the

city by crossing each bridge exactly once. The walk need not start and end

at the same part.

Seven bridges of Knigsberg

Euler1 approached this problem by representing the areas of land separated

by the river with vertices and represented the bridges with the edges joining

the vertices. The corresponding figure is as follows

which defines the graph of Knigsbergs 7 bridges.

1Leonhard Euler (1707-1782), Swiss mathematician.

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The problem now is to draw the above figure without retracing any line and

without picking the pencil up off the paper. In terms of graphs, to find a trail

which contains all the edges of the graph.

All vertices in the above picture are odd. Pick any of these, say the one with

degree 3. The first time we reach this vertex by an edge, we can leave by

another edge. But the next time we arrive, there is no edge by which we can

leave. So this vertex cannot be an intermediate vertex; it can only be the first

or the last vertex of the walk. Thus, an intermediate vertex cannot be odd so

it is impossible to draw the above graph in one pencil stroke without

retracing.

60-Eulerian trail. A trail which contains all the edges of a graph is called anEulerian trailand a closed such trail is called an Eulerian tour. A graph which

admits an Eulerian tour is called and a graph which admits anEulerian trail is said to be traceable. Above discussion shows that a for

graph to be traceable, two of the vertices must be odd and all the vertices

must be even. For the graph be Eulerian, all the vertices must be even. This

necessary condition is also sufficient:

THEOREM. A connected graph is Eulerian if and only if every vertex is even.Proof. Since the tour enters a vertex through some edge and leaves byanother edge, necessity of the condition is obvious. To show the sufficiency,

start with a vertex and begin making a tour. Keep going, never using thesame edge twice, until it is not possible to go further. Since every vertex is

even, the end point of the tour is . If all the edges are used, proof iscompleted. Otherwise each component of the subgraph consisting of unused

edges is a graph whose all vertices are even. Apply the same procedure to a

component to obtain a second tour. If this tour starts in a vertex of the first

tour, the two tours can be combined to produce a new tour. Continuing in

this manner one obtains a tour which includes all the edges.

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COROLLARY. A connected graph is traceable if and only if it has exactly twoodd vertices. In this case end points of the trail are he odd vertices.

THEOREM. A strongly regular connected digraph is Eulerian if and only if thein-valency of each vertex is equal to the out-valency.

Example. Among all platonic graphs, only the octahedron is Eulerian. is Eulerian if and only if is odd ( is Eulerian if and only if and are both even(

Example. Some Eulerian graphs.

Example. Some traceable graphs (odd vertices emphasized):

61-Hamiltonian2 graph. A path is called a Hamiltonian pathif it contains all thevertices, and similarly, a Hamiltonian cycle is a cycle which contains all the

vertices. A graph which admits a Hamiltonian cycle is said to be Hamiltonian.

In the mid 19-th century, Hamilton tried to popularize the exercise of finding

such a cycle in the dodecahedron. Below figure shows that dodecahedron isindeed a Hamiltonian graph.

2Named after Sir William Rowan Hamilton (1805-1865), Irish mathematician.

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62-We have seen that it can be easily checked whether a graph is Eulerian ornot. Moreover it is easy to construct an Eulerian trail (or tour) if there exists

any. In contrast to this, no trivial necessary and sufficient condition for a

graph to be Hamiltonian is known; the problem of deciding whether an

arbitrary graph admits a Hamiltonian cycle is hard3. We only have some

necessity and some sufficiency conditions.

Example. The graph given below is not Hamiltonian. To see this, notice thatthe graph is bipartite and has an odd number of vertices. But a bipartite

graph does not admit an odd cycle, hence the given graph does not admit a

cycle containing all the vertices.

In general if is odd, then (and of course any subgraph of it) is notHamiltonian.

63-THEOREM. If is Hamiltonian then, for every nonempty proper subset ||Proof. Let be a Hamiltonian cycle in . Then for every nonempty propersubset of we have ||. But is a spanning subgraph ofSo and theorem follows.

3In fact, the problem has been proved to be complete.

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Example. The following graph is bipartite and has an even number ofvertices. Thus, we cannot immediately decide whether it is Hamiltonian ornot. Now consider the set of white vertices Since and||

, we can conclude that the graph is not Hamiltonian.

Example. Petersen graph is not Hamiltonian but we cannot deduce this byusing above theorem.

64-THEOREM(Dirac). If is a simple graph with and then isHamiltonian.

65-THEOREM (Ore). If is a simple graph with and let are verticessuch that

Then

is Hamiltonian if and only if

is

Hamiltonian.

66-Closure. For a given graph with vertices, define inductively a sequence of graphs such that and for where and are nonadjacent vertices of such that .This procedure stops when no new edges can be added to for some Theresult of this procedure is the closureof, and its is denoted by In each step of the construction of

there are usually alternatives which

edge is to be added to the graph, and therefore the above procedure is notdeterministic. However, the final resultis independent of the choices. Agraph for which is called closed.

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Example. A graph and its closure

G Cl(G)

67-THEOREM. A simple graph with is Hamiltonian if and only if isHamiltonian.

68-COROLLARY. Let be a graph of order . If is the complete graph,then is Hamiltonian.

EXERCISES

VIII.1 Show that if has a Hamiltonian path then, for every proper subset of, || VIII.2. A mouse eats his way through a cube of cheese by tunnelingthrough all of the 27 unit subcubes. If he starts at one corner and always

moves on to an uneaten subcube, can he finish at the center of the cube?

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IX. PLANARITY

69-Planar and plane graphs. A graph can be represented in plane in arbitrarilymany different ways. If it is possible to represent a graph in the plane such

that no two edges intersect (except at vertices), we say that the graph is

planar(or embeddable in plane). Such a representation of a planar graph is

called aplane graph(or an embedding of the graph in plane).

Example. Below figure shows three representations of in plane. The firstone is not an embedding whereas the others are embeddings (plane graphs).

Not an embedding Embeddings

When we deal with planarity, parallel edges and loops are immaterial. In fact

a multi-graph is planar if and only if its underlying simple graph is planar. As

well, vertices of degree 1 or 2 do not have any effect on a graph for being

planar or not. We can summarize these as

PROPOSITION. A multi graph is planar if and only its underlying graph isplanar.

PROPOSITION. Let and be homeomorphic graphs. Then is planar if andonly if is planar.

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70-Face of a planar graph Eulers Polyhedral Formula. A plane graph partitionsthe rest of the plane into a number of bounded regions together with an

unbounded region. Each of these regions is called a faceof the plane graph.

The unbounded face is called the exterior face. The set and the number of

faces of a plane graph will be denoted respectively by and (or).exterior face

faces

Example. Aplane graph on 7 vertices with 14 edges and 9 faces.

PROPOSITION. Let be a plane graph,a) Two distinct faces of are disjoint; their boundaries can intersectonly on edges and vertices.

b) has a unique exterior face,c) A bridge belongs to the boundary of one face,d) Any edge which is not a bridge belongs to the boundary of two faces,e) Each cycle of

surrounds at least one internal face,

f) has no interior face if and only if it is acyclic (that is, has no cycles.)

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THEOREM. (Eulers Polyhedral Formula) Let be a connected plane graphwith vertices, edges and faces. Then Proof We proceed by induction on, the number of faces. If the onlyface is the unbounded face and the graph have no cycles. Such a graph is

connected if an only if it has edges. Then it follows that and the theorem holds. Now assume that the claim istrue for all connected plane graphs with less than and let be a gaph with faces. Pick some edge of G which is not a bridge. Then, is aconnected with

vertices,

edges and

faces. By

the induction hypothesis which implies that or . Let be a connected planar graph on vertices with edges and also letand be two different plane representations of. Let the number of facesof and be respectively, and. Since each of these plane graphs have

vertices and

edges Eulers polyhedral theore says that

and which implies that This observation means that,no matter how a planar graph is embedded in the plane, the number of faces

will always be the same. We call this common number of faces as the

number of faces of the graph itself.

COROLLARY Let be a graph with vertices, edges and faces and

components. Then

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71-Dual of a Plane Graph. Given a plane graph , we define the dual graph tobe new plane graph as follows. Each vertex of corresponds to a face ofand each edge of corresponds to an edge of . Two vertices of arejoined if and only if the corresponding faces are incident in . To each edgein the separating two faces in there is an edge joining the correspondingvertices in Example.

G Each edge of corresponds toan edge of

It should be noted that dual of any plane graph is connected even if the graph

itself is disconnected. It follows that a disconnected plane graph cannot be

isomorphic to the dual of the dual graph. On the other hand, it follows from

the construction that dual of dual is isomorphic to the graph itself.

THEOREM. If is a connected plane graph then

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Let be a connected plane graph with parameters and be the dualwith parameters . From the definition it follows that and Since both graphs are connected plane graphs Eulers polyhedralformula implies that Isomporphic plane graphs do not necessarily have isomorphic duals.Example. The following graphs and are isomorphic, but their duals and are not isomorphic.

72-Maximal Planar Graphs

Each face of a plane graph is an open connected subset of the plane and

boundary of a face consists of edges and vertices of the graph. For any face ,

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the number of edges in the boundary is called the degree of the face,

denoted by . A bridge in a face contributes 2 to the degree of that face.A face is said to be incidentwith the edges and faces in its boundary. Any

edge is incident with exactly two edges unless it is a bridge. A bridge isincident with only one face. So, when the degrees of faces are added, each

edge (even a bridge), is counted twice and consequentely, the sum of

degrees of faces is twice the number of edges. Recaling that sum of local

degrees of vertices is also twice the number of edges we have

THEOREM. Let be a plane graph where and are the set of vertices andlet the number of edges be

, then

A simple planar graph is said to be maximal planar if it contains largestpossible number of edges. In other words, if whenever any pair of non-

adjacent vertices are joined by an edge, the resulting graph is non-planar,

then we say that is maximal planar. In a similar manner, we say that agraph

is minimal non-planarif it is non-planar and

is planar for any

edge of . If a face of a simple planar graph has degree larger than 3, we candraw a new edge in this face. This means that a maximal planar graph cannothave a face with degree 4 or more. Consequently, all the faces of a maximal

planar graph are triangles. Such a graph is also called a triangulation.

A maximal plane graph on 7 vertices with 15 edges and 10 faces.

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THEOREM. Let be a simple planar connected graph on vertices with edges. If all the faces have degree 3, then Proof. If each face is a triangle, we get

Substituting

in we obtain COROLLARY 1. Let be a simple planar connected graph on vertices with edges. Then COROLLARY 2. Let be a simple planar connected graph on vertices with edges. Let be the girth of Then, COROLLARY 3. Let be a simple planar connected graph on vertices with edges. If has no triangles, then THEOREM. For any simple planar graph , .Proof. Without loss of generality we may assume that is connected and . Then and since we get 6 , that is 6 Thus,

73-Theorem of Kurtowski

THEOREM. and are non-planar.For , and . By the first corollary to the theorem,

cannot be planar.

For , 6 9 and . Since has no triangles, bycorollary 3 we conclude that it is non-planar.

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Converse of above theorem is also true and this famous result is known asKuratowskis theoreTHEOREM. (Kuratowski, 1930) A graph is planar if and only if it contains nosubgraph homeomorphic to or .Example. Any complete graph on 5 or more vertices is nonplanar.

Example. In the below figure a subgraph of Petersen graph which ishomeomorphic to is shown. Thus, Petersen graph is not planar.

In terms of contractions, a useful extension of Kuratowskis theore is givenby Wagner:

THEOREM. (Wagner, 1937) A graph is planar if and only if it contains nosubgraph contractible to or .Example. Considering the contractions along the emphasized edges in thebelow figure, it is seen that the Petersen graph is contractible to . Oncemore we see that Petersen graph is nonplanar.

74-Crossing Number. In a representation of a graph in plane, the total numberof crossings of pairs of edges (other than vertices) is called the crossingnumber of the representation. Among all representations of in plane, the

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minimum value of the crossing number is defined to be the crossing number of the graph. For a planar graph obviously

Example. Below representations of and show that these graphs havea crossing number of 1. In a sense, a chromatic number measures how non-

planar is a given graph.

Conjecture (Guy, 1972). Crossing number of the complete graph is The conjecture is known to be true for For the larger values, theconjectured crossing number is known to be an upper bound.

Conjecture (Zarankiewicz, 1954). Crossing number of the complete bipartite

graph is()

It was shown that for the complete bipartite and the crossingnumbers are exactly as given

by Zarankiewiczs conjecture Kletian 9.

In the general case, the conjectured value is known to be an upper bound.

http://en.wikipedia.org/wiki/Kazimierz_Zarankiewiczhttp://en.wikipedia.org/wiki/Kazimierz_Zarankiewicz

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In general, determining the crossing number of a graph is a hard problem. In

fact it was shown that it is an NP-hard proble (Garey and Johnson, 1983).

Two known bounds for the chromatic number are as follows:

Theorem. For any graph on vertices with edgesa) b)

EXERCISES

IX.1 Show that if , then IX.2 Letbe a maximal simple planar graph (triangulation). Show that is a

2-edge connected 3-regular planr graph.

IX.3 If is a simple planar graph on edges, then show that thecomplement of

is non-planar.