Grah Theory

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Topics in Discrete MathematicsIntroduction to Graph Theory

Graeme Taylor

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DenitionAn embedding divides the plane into some number f of regions orfaces ; one of these has innite area, and is called the inniteface . For each face F , we dene its degree deg (F ) to be thenumber of edges encountered in a walk around the boundary of F .

A B C D

For this embedding, face A has degree three, B has degree four, C has degree ve - not three! - and D , the innite face, has degreesix.


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DenitionFor any n N , the complete graph K n is the n-vertex graph inwhich every vertex is adjacent to every other.

K 1 K 2 K 3 K 4 K 5


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DenitionA graph G = ( V , E ) is bipartite if V can be divided into disjointsets V 1, V 2 such that every edge of G is of the form {a , b } witha V 1, b V 2.If every vertex of V 1 is joined to every vertex of V 2, with |V 1 | = m,|V 2 | = n, then G is the complete bipartite graph K m , n .

K 1, 12 K 3 , 3


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Theorem (Eulers Formula)

Let G = (

V , E ) be a connected planar graph such that

|V

| = v ,

|E | = e. Suppose there is an embedding of G with f faces. Then

v e + f = 2 .

We will prove this by strong induction on e . For e = 0,

A

For e = 1 A


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Let k N , and assume the result is true for any connected planargraph with e edges where 0 e k .Let G be a connected planar graph with v vertices and e = k + 1edges embedded so as to give f faces.Delete an edge {a , b } E to give the subgraph H = G {a , b }.H is either connected or disconnected.If H is connected:

Neither vertex a or b can have degree one in G . So {a , b } acts as the boundary between two faces in the

embedding of G , which merge in H . H therefore has v vertices, k = e 1 edges and f 1 faces.

By the induction hypothesis,2 = v (e 1) + ( f 1) = v e + f .


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Let k N , and assume the result is true for any connected planargraph with e edges where 0 e k .Let G be a connected planar graph with v vertices and e = k + 1

edges embedded so as to give f faces.Delete an edge {a , b } E to give the subgraph H = G {a , b }.H is either connected or disconnected.If H is disconnected:

H has two components H 1, H 2 with v 1 , e 1 , f 1 and v 2 , e 2, f 2vertices, edges and faces respectively.

We know v 1 + v 2 = v , e 1 + e 2 = k = e 1 andf 1 + f 2 = f + 1 due to double-counting of the innite face.

Applying the inductive hypothesis to each of H 1 , H 2 we have

v 1 e 1 + f 1 = 2 and v 2 e 2 + f 2 = 2

so 4 = (v 1 + v 2) (e 1 + e 2) + ( f 1 + f 2) =v (e 1) + ( f + 1) = v e + f + 2, i.e., v e + f = 2.


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Lemma (Handshaking lemma for planar graphs)

For a connected planar graph G,

i

deg (F i ) = 2 |E | .

Each edge either borders two faces - and so contributes once to

the degree of each of them - or terminates in a vertex of degreeone inside some face, and thus is encountered twice in the walkaround that face.

A B C D


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CorollaryLet G = ( V , E ) be a connected planar graph such that |V | = v ,|E | = e > 2. Suppose there is an embedding of G with f faces.Then 2e 3f and e 3v 6.

The boundary of each face contains at least three edges, so

2e =f

i =1

deg (F i ) 3f .

By Eulers Formula, 2 = v e + f , so 2 v e + 23 e = v e

3 , so6 3v e , or e 3v 6.


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The graph K n has n vertices and n (n 1)2 edges. Thus K n is nonplanar for any n 5.


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CorollaryLet G = ( V , E ) be a connected bipartite planar graph such that |V | = v , |E | = e > 2. Suppose there is an embedding of G with f faces. Then 2e 4f and e 2v 4.

Thus the complete bipartite graph K 3, 3 - with 6 vertices, and

9 edges - cannot be planar.


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DenitionLet G = ( V , E ) be a graph such that E = . Pick an edgee = {u , w } V , add a new vertex v to V and replace e with newedges {u , v } and {v , w }. The resulting graph is called anelementary subdivision of G .

DenitionTwo graphs are homeomorphic if they are isomorphic, or they canboth be obtained from the same starting graph H by a sequence of

elementary subdivisions.


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Theorem (Kuratowksis Theorem)

A graph is nonplanar if and ony if it contains a subgraph that is homeomorphic to either K 5 or K 3 , 3.


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Lemma

If G is obtained from H by a sequence of elementary subdivisions,then G is planar if and only if H is.

Suppose H is planar. Then we may take an embedding of H , andthen recover G by following the sequence of subdivisions, adding

new vertices along the midpoint of the edge being replaced. Butthen this is an embedding of G .Conversely, suppose H is nonplanar. Then G cannot be planar, forif it were, we could take an embedding of G , then reverse thesequence of subdivisions, each time deleting a vertex of degree 2and merging the two edges incident at it, ultimately arriving at anembedding of H .


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Example

The graph

is non-planar.


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G G K 3, 3


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ExampleThe graphs

both have characteristic polynomialx 7 13x 5 12x 4 + 17 x 3 + 10 x 2 8x

and are therefore cospectral.So the spectrum cannot determine planarity!


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DenitionA regular graph is one in which every vertex has the same degree.If that degree, d , is known, we call it a d -regular graph. A3-regular graph may also be described as cubic .

DenitionA regular solid is a solid geometric gure where each face is thesame regular polygon (a polygon with all angles equal, and allsides of the same length), and the same number of such facesmeet at each corner (vertex).


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TheoremThe only regular solids, are the ve Platonic solids : the tetrahedron, cube, octahedron, dodecahedron and icosahedron.


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Tetrahedron


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Cube


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Icosahedron


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The graph of the vertices and edges of a regular solid is planar

and d -regular for some d 3. By handshaking, dv = 2e . Each face has the same degree k 3, so kf = 2e by

handshaking for planar graphs. Thus

v = 2e

d and f = 2e

k . By Eulers Formula, v e + f = 2. So

1d

+ 1k

= 1e

+ 12

>12

.


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In molecular chemistry, a fullerene is an allotrope of carbon thattakes the form of a hollow sphere, ellipsoid or tube.

DenitionA fullerene is a 3-regular planar graph in which every face of its

embedding has degree ve or six.

PropositionA fullerene has exactly twelve faces of degree ve.

Proof


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Suppose G is a fullerene with v vertices, e edges, f 5 faces of degree 5, and f 6 faces of degree 6 (so G has f = f 5 + f 6 faces).

By Eulers formula, f = 2 + e v so

f 5 + f 6 = 2 + 3n

2 v = 2 +

v 2

.

By standard handshaking, the 3-regularity of G gives 2e = 3v ;and by handhaking for planar graphs we have 2e = 5 f 5 + 6 f 6,so

3v = 5 f 5 + 6 f 6 = 5( f 5 + f 6) + f 6 = 5(2 + v 2

) + f 6.

From this we obtain v = 2 f 6 + 20, and so

f 5 = 2 + v 2

f 6 = 2 + v 2

(v 2

10) = 12 .


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Example

The Dodecahedron is the smallest possible fullerene.

Denition


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An isolated pentagon fullerene is one in which no twopentagonal faces share a common vertex.

Example

A 60-vertex isolated pentagon fullerene.


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Buckminsterfullerine , The buckyball.


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The Montreal Biosphere, one of Buckminster Fullers geodesicdomes.

Grah Theory

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