CSCI 4260MATH 4150

1
CSCI 4260-MATH 4150

• Embeddings

2
Embedding graphs on surfaces

• Last lecture
• Graph G is planar G can be drawn on the plane in
  a way that no two edges cross.
• This drawing is called an embedding of G in the
  plane
• Today
• Embeddings of non-planar graphs
• If G is non-planar, it can not be embedded in the
  plane

3
Intuition

• Lets look at K(5), which we know is non-planar

4
The main idea

• We will start with a basic surface
• Add handles as necessary
• The basic surface can be the plane
• But it is easier to visualize certain properties
  on the sphere
• In fact, in terms of embedability the plane and
  sphere is equivalent

5
Plane-sphere equivalence

• Embedding on the plane ? embedding on the sphere
• How about the other direction
• Can you convert an embedding on the sphere to an
  embedding on the plane

6
From http//www.heartofmath.com/first_edition/pdf
s/pg360.pdf
7
Torus a sphere with a handle
8
Constructing a torus

• http//www.cmis.brighton.ac.uk/jt40/MapleAnimatio
  ns/Torus.html
• Local copy

9
Back to K(5)
10
Surface genus number of handles

• k-handles
• Surface of genus k
• S(k)
• S(0) sphere
• S(1) torus

11
Embedding graphs

• Given G, draw it on the sphere
• If two edges cross, add a handle
• Note we can draw the graph in a way that only
  two edges cross at any intersection

12
Genus of a graph G

• The smallest k such that G can be embedded on
  S(k)
• Lets see if/how the properties of planar graphs
  generalize

13
Remember Euler Equation

• n-mr 2 for connected, planar graphs
• In other words, number of regions is the same for
  any planar embedding

14
Three embeddings of K(4) on a torus
ERROR IN THE BOOK
? regions
4 regions
? regions
15
Number of regions

• Depends on the embedding
• The problem is that there is too much variety
• We need the notion of a canonical embedding

16
2-cells

• A region is called a 2-cell if any closed curve
  that is drawn inside the region can be
  continuously shrunk to a single point

17
Euler Equation
R2
R1
n 3, m 3, r 2 n-mr 2 Both R1 and R2 are
2-cell. Why?
18
Euler Equation for diconnected planar graphs?
R3
R1
R2
n 6, m 6, r 3 n-mr 3 the equation is not
true. Is R3 a 2-cell region?
19
Which of these embeddings of K(4) is 2-cell?
20
Theorem

• Let G be a connected graph that is 2-cell
  embedded on a surface of genus k. Then,
• n m r 2 2k
• where n orderm orderr number of regions

21
Proof (by induction on k)

• Basis k 0.
• G must be planar
• n m r 2 2k
• n m r 2
• Eulers equation

22
Inductive step

• Suppose the theorem is true for all graphs that
  can be 2-cell embedded on surfaces with k or less
  handles
• Let G be a graph 2-cell embedded on S(k1)
• Let H be one of the handles

23
A closer look at H

• We can always embed in a way that there are no
  vertices on handles
• Case 1 there are no edges on the handle.
• (ind. hyp. holds)
• Now suppose there are edges along the handle

24
A closer look at H

• Draw a curve C around H
• Let t be the of intersections with edges
• Insert a new vertex at every intersection
• Insert two more vertices to prevent parallel
  edges or loops

C
25
We now a have a new graph

• G1
• n1 n t 2
• m1 m 2t 2
• r1 r t

C
26
Now cut the handle

• G1
• n1 n t 2
• m1 m 2t 2
• r1 r t
• G2
• n2 n1 t 2
• m2 m1 t 2
• r2 r1 2

27
We can now apply the inductive hypothesis

• (n2t4) (m3t4) (r t 2) 2 2k
• n m r 2 2 - 2k
• n m r 2 2(k1)
• done.

28
Recall the genus of a graph

• ??(G) the min. k such that G can be embedded in
  S(k)
• Theorem (w/o proof)
• If we embed G on S(?(G)), the resulting embedding
  is 2-cell.

29
Corollary

• ? Let G be a connected graph that is embedded on
  a surface of genus ?(G). Then,
• n m r 2 2 ?(G)

30
Recall

• For planar graphs we have m 3n 6
• Similarly, we have

31
Corollary

• If G is a connected graph of order n 3 and size
  m, then

32
Proof

• Similar to the planar case, we need to get rid of
  r from the equation
• n m r 2 2 ?(G)
• Let
• R1,, Rr be the regions, and
• m1,,mr be the number of edges on the boundaries
  of respective regions.

33
Proof (cont)

• We have
• mi 3
• Every edge is counted at most twice
• Which gives us 3r 2m

34
Using 3r 2m

• 2 2 ?(G) n m r
• 6 6 ?(G) 3n 3m 3r 3n 3m 2m
  3n m
• Rearranging, we get

35
A final remark

• Recall the argument about starting with a
  embedding on the sphere and adding handles at the
  intersection
• This suggests that the genus of a graph of order
  m is O(n2)
• Can it ?be ?(n2)?

36
Theorem
37
A note

• I am traveling this week until Sunday
• No office hour on Thursday
• I am maybe slow in responding to email until
  Sunday Night
• Nikhil will hold his regular office hours and
  review matchings on Thursday lecture.