Topics in Discrete MathematicsIntroduction to Graph Theory

Graeme Taylor

4/ii/13

In this section, we’ll try to reintroduce some geometry to our studyof graphs.

Definition

A planar graph is one which can be drawn in the plane withoutany edges crossing. Such a drawing is called an embedding of thegraph in the plane.

Definition

An embedding divides the plane into some number f of regions orfaces; one of these has infinite area, and is called the infiniteface. For each face F , we define 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, Chas degree five - not three! - and D, the infinite face, has degreesix.

Definition

An embedding divides the plane into some number f of regions orfaces; one of these has infinite area, and is called the infiniteface. For each face F , we define 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 of the same graph we have two faces of degreethree, one of degree four, and one of degree eight:The list of face degrees is not a graph invariant!

Definition

For any n ∈ N, the complete graph Kn is the n-vertex graph inwhich every vertex is adjacent to every other.

K1 K2 K3 K4 K5

Definition

A graph G = (V ,E ) is bipartite if V can be divided into disjointsets V1,V2 such that every edge of G is of the form {a, b} witha ∈ V1, b ∈ V2.If every vertex of V1 is joined to every vertex of V2, with |V1| = m,|V2| = n, then G is the complete bipartite graph Km,n.

K1,12 K3,3

Theorem (Euler’s 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 = 1A

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 theembedding 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 .

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 disconnected:

• H has two components H1, H2 with v1, e1, f1 and v2, e2, f2vertices, edges and faces respectively.

• We know v1 + v2 = v , e1 + e2 = k = e − 1 andf1 + f2 = f + 1 due to double-counting of the infinite face.

• Applying the inductive hypothesis to each of H1,H2 we have

v1 − e1 + f1 = 2 and v2 − e2 + f2 = 2

so 4 = (v1 + v2)− (e1 + e2) + (f1 + f2) =v − (e − 1) + (f + 1) = v − e + f + 2, i.e., v − e + f = 2.

Lemma (Handshaking lemma for planar graphs)

For a connected planar graph G ,∑i

deg(Fi ) = 2|E |.

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

Corollary

Let 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(Fi ) ≥ 3f .

By Euler’s Formula, 2 = v − e + f , so 2 ≤ v − e + 23e = v − e

3 , so6 ≤ 3v − e, or e ≤ 3v − 6.

• The graph Kn has n vertices and n(n−1)2 edges.

• Thus Kn is nonplanar for any n ≥ 5.

Corollary

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

• Thus the complete bipartite graph K3,3 - with 6 vertices, and9 edges - cannot be planar.

Definition

Let G = (V ,E ) be a graph such that E 6= ∅. 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 .

Definition

Two graphs are homeomorphic if they are isomorphic, or they canboth be obtained from the same starting graph H by a sequence ofelementary subdivisions.

Theorem (Kuratowksi’s Theorem)

A graph is nonplanar if and ony if it contains a subgraph that ishomeomorphic to either K5 or K3,3.

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, addingnew 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.

We therefore have the reverse direction of Kuratowski’s Theorem:

• Being planar is monotone decreasing - if we can embed G , wecan embed any subgraph obtained by deleting edges of G .

• So if G has a non-planar subgraph then G cannot be planar.

• By the previous lemma, if two graphs are homeomorphic, thenthey are both planar, or both non-planar.

• So if G has a subgraph G ′ homeomorphic to K5 or K3,3, thenG ′ is non-planar, and hence so is G .

Example

The graph

is non-planar.

G G ′ K3,3

Example

The graphs

both have characteristic polynomial

x7 − 13x5 − 12x4 + 17x3 + 10x2 − 8x

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

Definition

A 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.

Definition

A regular solid is a solid geometric figure 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).

Theorem

The only regular solids, are the five Platonic solids: thetetrahedron, cube, octahedron, dodecahedron and icosahedron.

Tetrahedron

Octahedron

Cube

Icosahedron

Dodecahedron

• The graph of the vertices and edges of a regular solid is planarand d-regular for some d ≥ 3. By handshaking, dv = 2e.

• Each face has the same degree k ≥ 3, so kf = 2e byhandshaking for planar graphs.

• Thus

v =2e

dand f =

2e

k.

• By Euler’s Formula, v − e + f = 2. So

1

d+

1

k=

1

e+

1

2>

1

2.

1

d+

1

k=

1

e+

1

2>

1

2.

If d ≥ 4 and k ≥ 4 then the LHS is at most 12 . So at least one of

d , k is at most 3. But both are also at least 3. So either:

1 d = 3, k ≥ 3: then 13 + 1

k = 1e + 1

2 so 1k = 1

e + 16 ≥

16 so

k = 3, 4, or 5;

2 k = 3, d ≥ 3: then d = 3, 4 or 5.

That is, (d , k) ∈ {(3, 3), (3, 4), (3, 5), (4, 3), (5, 3)}, giving the fivecases of the tetrahedron, cube, dodecahedron, octahedron andicosahedron.

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

Definition

A fullerene is a 3-regular planar graph in which every face of itsembedding has degree five or six.

Proposition

A fullerene has exactly twelve faces of degree five.

Proof

• Suppose G is a fullerene with v vertices, e edges, f5 faces ofdegree 5, and f6 faces of degree 6 (so G has f = f5 + f6 faces).

• By Euler’s formula, f = 2 + e − v so

f5 + f6 = 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 = 5f5 + 6f6,so

3v = 5f5 + 6f6 = 5(f5 + f6) + f6 = 5(2 +v

2) + f6.

• From this we obtain v = 2f6 + 20, and so

f5 = 2 +v

2− f6 = 2 +

v

2− (

v

2− 10) = 12.

Example

The Dodecahedron is the smallest possible fullerene.

Definition

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

Example

A 60-vertex isolated pentagon fullerene.

Buckminsterfullerine, The ‘buckyball’.

The Montreal Biosphere, one of Buckminster Fuller’s geodesicdomes.