(1/27)

MA284 : Discrete MathematicsWeek 9: Planar and non-planar graphs

http://www.maths.nuigalway.ie/˜niall/MA284/

7 and 9 November, 2018

1 Planar graphs and Euler’s formula2 Non-planar graphs

K5K3,3Every other non-planar graph

3 PolyhedraEuler’s formula for convex polyhedra

4 How many Platonic solids are there?5 Exercises See also Section 4.2 of Levin’s

Discrete Mathematics: an openintroduction.

Wednesday's slides

Announcement (2/27)

ASSIGNMENT 3: deadline today (Wednesday, 7November) 5pm.

ASSIGNMENT 4: deadline Friday, 16 November at5pm.

Planar graphs and Euler’s formula (3/27)

Recall: Planar graphsIf you can sketch a graph so that none of its edges cross, then it is aPLANAR graph.When a planar graph is drawn without edges crossing, the edges andvertices of the graph divide the plane into regions. We will call eachregion a FACE . The “exterior” of the graph is consider a face.

Planar graphs and Euler’s formula (4/27)

Towards the end of last Friday’s class, we produced a list of planar graphsand counted the vertices, edges, and faces. They all had something incommon...

Euler’s formula for planar graphsFor any (connected) planar graph with v vertices, e edges andf faces, we have

v − e + f = 2

Outline of proof:

Planar graphs and Euler’s formula (5/27)

(Proof continued).

Planar graphs and Euler’s formula (6/27)

ExampleApplication of Euler’s formula Is it possible for a planar graph to have 5vertices, 7 edges and 3 faces? Explain.

Non-planar graphs (7/27)

Of course, most graphs do not have a planar representation. We havealready met two that (we think) cannot be drawn so no edges cross: K5and K33:

However, it takes a little work to prove that these are non-planar. While,through trial and error, we can convince ourselves these graphs are notplanar, a proof is still required.

For this, we can use Euler’s formula for planar graphs to prove theyare not planar.

Non-planar graphs K5 (8/27)

Theorem: K5 is not planar.The proof is by contradiction:

Non-planar graphs K3,3 (9/27)

Theorem (K3,3 is not planar)This is Theorem 4.2.2 in the text-book. Please read the proof there.

The proof for K3,3 is somewhat similar to that for K5, but also uses thefact that a bipartite graph has no 3-edge cycles...

Non-planar graphs Every other non-planar graph (10/27)

The understand the importance of K5 and K3,3, we first need the conceptof homeomorphic graphs.

Recall that a graph G1 is a subgraph of G if it can be obtained bydeleting some vertices and/or edges of G .

A SUBDIVISION of an edge is obtained by “adding” a new vertex ofdegree 2 to the middle of the edge.

A SUBDIVISION of a graph is obtained by subdividing one or more of itsedges.

Example:

Non-planar graphs Every other non-planar graph (11/27)

Closely related: SMOOTHING of the pair of edges {a, b} and {b, c},where b is a vertex of degree 2, means to remove these two edges, andadd {a, c}.

Example:

Non-planar graphs Every other non-planar graph (12/27)

The graphs G1 and G2 are HOMEOMORPHIC if there is somesubdivision of G1 is isomorphic to some subdivision of G2.

Examples:

Non-planar graphs Every other non-planar graph (13/27)

There is a celebrated theorem due to Kazimierz Kuratowski. The proof isbeyond what we can cover in this module. But if you are interested inMathematics, read up in it: it really is a fascinating result.

Theorem (Kuratowski’s theorem)A graph is planar if and only if it does not contain a subgraph that ishomeomorphic to K5 or K3,3.

What this really means is that every non-planar graph has somesmoothing that contains a copy of K5 or K3,3 somewhere inside it.

ExampleThe Petersen graph is not planar https://upload.wikimedia.org/wikipedia/commons/0/0d/Kuratowski.gif

Finshed here Wedneday