1. many branches of mathematics
Graphs and polyhedra:
From Euler to
many branches of mathematics
László Lovász
Eötvös University, Budapest

2. Chinese Postman Problem
Euler and graph theory
The Königsberg bridges
Eulerian graphs
Chinese Postman Problem

3. Traveling Salesman Problem
Euler and graph theory
The Knight’s Tour
Hamilton cycles
Traveling Salesman Problem
P vs. NP-complete

4. Euler and graph theory
The Polyhedron theorem

5. combinatorial structure!
Euler and graph theory
The Polyhedron theorem
#vertices - #edges + #faces = 2
Polyhedra have
combinatorial structure!
algebraic topology (Euler characteristic)
combinatorics of polyhedra
Möbius function
...

6. For every planar graph, #edges ≤ 3 #nodes - 6
Convex polyhedra and planar graphs
3-connected
planar graph
For every planar graph, #edges ≤ 3 #nodes - 6

7. Every planar graph can be drawn in the plane with straight edges
Planar graphs: straight line representation
planar graph
Every planar graph can be drawn
in the plane with straight edges
Fáry-Wagner

8. 3-connected planar graph
Planar graphs and convex polyhedra
3-connected planar graph
Steinitz 1922
Every 3-connected planar graph
is the skeleton of a convex 3-polytope.

9. Every 3-connected planar graph can be drawn with
Rubber band representation
Tutte (1963)
Every 3-connected planar graph can be drawn with
straight edges and convex faces.
outer face fixed to
convex polygon
edges replaced by
rubber bands
Discrete harmonic and analytic functions
Energy:
Equilibrium:

10. rubber band embedding is planar
Rubber band representation
G 3-connected planar
rubber band embedding is planar
(Easily) polynomial time
computable
Lifts to Steinitz
representation if
outer face is a triangle
Maxwell-Cremona
Tutte
Demo!

11. Discrete version of the Riemann Mapping Theorem
Coin representation
Koebe (1936)
Discrete version of the Riemann Mapping Theorem
Every planar graph can be represented by touching circles

12. # ≤ #faces (Euler)
= #edges - #nodes + 2
≤ 2 #nodes - 4
< 2 #nodes

13. Every 3-connected planar graph is the skeleton of a convex polytope
Coin representation
Polyhedral version
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
Andre’ev

14. Coin representation
From polyhedra to circles
horizon

15. Coin representation
From polyhedra to representation of the dual

16. Largest has multiplicity 1.
The Colin de Verdière number
G: connected graph
Roughly: multiplicity of second largest eigenvalue
of adjacency matrix
Largest has multiplicity 1.
But: maximize over weighting the edges and diagonal entries
But: non-degeneracy condition on weightings

17.  M=(Mij): symmetric VxV matrix <0, if ijE Mii arbitrary Mij 0, if
The Colin de Verdière number
Formal definition
M=(Mij): symmetric VxV matrix
Mij
<0, if ijE
0, if
Mii arbitrary
normalization
M has =1 negative eigenvalue
Dimension of solutions of certain PDE’s
symmetric, 
X=0
Strong Arnold Property

18. for μ>2, μ(G) is invariant under subdivision
The Colin de Verdière number
Basic Properties
μ≤k is polynomial time
decidable for fixed k
μ(G) is minor monotone
deleting and
contracting edges
for μ>2, μ(G) is invariant under subdivision

19. μ(G)≤2  G is outerplanar
The Colin de Verdière number
Special values
μ(G)≤1  G is a path
μ(G)≤2  G is outerplanar
μ(G)≤3  G is a planar
Colin de Verdière, using pde’s
Van der Holst, elementary proof
μ(G)≤4  G is linklessly embedable in 3-space …
μ(G)≥n-4  complement G is planar _ ~
Kotlov-L-Vempala

20. Representation of G in 
The Colin de Verdière number
Nullspace representation
basis of nullspace of M
Representation of G in 

21. Discrete version of Courant’s Nodal Theorem
The Colin de Verdière number
Van der Holst’s Lemma
connected
Discrete version of Courant’s Nodal Theorem
or…
like convex polytopes?

22.  G 3-connected nullspace representation gives planar
The Colin de Verdière number
Steinitz representation
G 3-connected
planar 
nullspace representation gives
planar embedding in 2
The vectors can be rescaled so that
we get a convex polytope.

23. Colin de Verdière matrix M Steinitz representation P
The Colin de Verdière number
Steinitz representation
Colin de Verdière matrix M
Steinitz representation P q p u v

24. ?  G path nullspace representation gives embedding in 1
The Colin de Verdière number
Nullspace representation III
G path 
nullspace representation gives
embedding in 1
G 2-connected
outerplanar 
nullspace representation gives
outerplanar embedding in 2
G 3-connected
planar 
nullspace representation gives
Steinitz representation ?
G 4-connected
linkless embed.
nullspace representation gives
linkless embedding in 3 