Planar Graphs

1 / 28 Planar Graphs

Definition A graph is planar if it can be drawn in the plane with no edges crossing.

2 / 28 Examples of Planar Graphs

C4 and K4 are both planar.

3 / 28 Proof Idea Remove edges until no cycles exist, always maintaining connectivity. Each removal of an edge from a cycle reduces the number of edges by 1 and the number of regions by 1, and leaves the number of vertices unchanged. The resulting graph is a tree, which has one less edge than vertices, and which has one region.

Euler’s Formula Theorem For any connected plane graph with p vertices and q edges that divides the plane into r regions, we have

p − q + r = 2.

4 / 28 Euler’s Formula Theorem For any connected plane graph with p vertices and q edges that divides the plane into r regions, we have

p − q + r = 2.

Proof Idea Remove edges until no cycles exist, always maintaining connectivity. Each removal of an edge from a cycle reduces the number of edges by 1 and the number of regions by 1, and leaves the number of vertices unchanged. The resulting graph is a tree, which has one less edge than vertices, and which has one region.

5 / 28 Regional Degree Theorem

Definition The degree of a region of a plane graph is the number of encounters with edges in a walk around the boundary of the region.

Theorem For any connected planar graph with q edges that divides the plane into r regions, we have

2q = sum of degrees of the regions .

6 / 28 Example of a Non-Planar Graph Consider a connected graph with p = 6 vertices and q = 13 edges. If the graph were planar, then by Euler’s formula it would have r = 9 regions.

c

b a f d

e

7 / 28 K5 is not planar

K5 has p = 5 vertices and q = 10 edges. If K5 were planar, it would have r = 7 regions.

b

a c

e d

8 / 28 K3,3 is not planar

K3,3 has p = 6 vertices and q = 9 edges. If K3,3 were planar, it would have r = 5 regions.

a b c

d e f

9 / 28 Petersen graph is not planar Petersen graph has p = 10 vertices and q = 15 edges. If Petersen graph were planar, it would have r = 7 regions.

g

h b c a f d e i

j

10 / 28 What makes a graph non-planar?

I Does having large cycles make a graph non-planar?

I Does having many edges make a graph non-planar?

I Does having a non-planar subgraph make a graph non-planar?

11 / 28 Planarity of Complete Graphs

Theorem Kn is planar if and only if n ≤ 4.

12 / 28 Planarity of Complete Bipartite Graphs

Theorem Km,n is planar if and only if min(m, n) ≤ 2.

13 / 28 Is this graph non-planar?

c

b a f d

e

14 / 28 Subdivision of graphs

Definition Inserting a new vertex into an existing edge of a graph is called subdividing the edge.

One or more subdivisions of edges creates a subdivision of the original graph.

15 / 28 Kuratowski’s Theorem

Theorem A graph is planar if and only if it does not contain a subdivision of K5 or K3,3 as a subgraph.

16 / 28 Planarity Algorithm for Hamiltonian Graphs

Let G be a Hamiltonian graph. 1. Draw the graph G with Hamilton cycle H on the outside, i.e. with H as the boundary of the infinite region.

2. List the edges of G not in H: e1,..., er . 3. Form a new graph K in which the vertices are labelled e1,..., er and where the vertices labelled ei , ej are joined by an edge if and only if ei , ej cross in the drawing of G, i.e. cannot both be drawn inside (or outside) H (such edges are said to be incompatible). 4. Then G is planar if and only if K is bipartite.

17 / 28 Example

Verify that the given graph has a Hamilton cycle and find a plane diagram for the graph.

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a

d e

18 / 28 Example

Hamilton cycle drawn on Corresponding incompatible the outside. edge graph.

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d ce bc b e

19 / 28 Convex Regular Polyhedra A polyhedron is convex when the straight line segment joining any two of its vertices lies entirely within it. A convex polyhedron can be represented by a planar graph.

A polyhedron is regular when there exist integers m ≥ 3, n ≥ 3 such that each vertex has m edges meeting at it, and each face has n edges on its boundary.

20 / 28 Tetrahedron

Edges: 6; Vertices: 4; Regions: 4

21 / 28 Cube

Edges: 12; Vertices: 8; Regions: 6

22 / 28 Octahedron

Edges: 12; Vertices: 6; Regions: 8

23 / 28 Dodecahedron

Edges: 30; Vertices: 20; Regions: 12

24 / 28 Icosahedron

Edges: 30; Vertices: 12; Regions: 20

25 / 28 Classification of Convex Regular Polyhedra

Theorem Suppose that a regular polyhedron has each vertex of degree m and each face of degree n.

Then (m, n) is one of (3, 3), (3, 4), (4, 3), (3, 5), (5, 3).

Further, there exist Platonic solids corresponding to each of these pairs.

26 / 28 Platonic Solids Corresponding to (m, n)

m n q p r Name 3 3 6 4 4 tetrahedron 3 4 12 8 6 cube 4 3 12 6 8 octahedron 3 5 30 20 12 dodecahedron 5 3 30 12 20 icosahedron

27 / 28 Acknowledgements

Statements of results follow the notation and wording of Anderson’s First Course in Discrete Mathematics.

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