Graphs and polyhedra:

From Euler to

many branches of mathematics

László Lovász

Eötvös University, Budapest

Euler and graph theory The Königsberg bridges

Eulerian graphsChinese Postman Problem

Euler and graph theory The Knight’s Tour

Hamilton cyclesTraveling Salesman Problem

P vs. NP-complete

Euler and graph theory The Polyhedron theorem

Euler and graph theory The Polyhedron theorem

#vertices - #edges + #faces = 2

algebraic topology (Euler characteristic)combinatorics of polyhedraMöbius function...

Polyhedra have combinatorial structure!

Convex polyhedra and planar graphs

3-connected planar graph

For every planar graph, #edges ≤ 3 #nodes - 6

Planar graphs: straight line representation

Every planar graph can be drawnin the plane with straight edges

Fáry-Wagner

planar graph

Steinitz 1922Every 3-connected planar graph

is the skeleton of a convex 3-polytope.

3-connected planar graph

Planar graphs and convex polyhedra

Every 3-connected planar graph can be drawn with straight edges and convex faces.

outer face fixed toconvex polygon

edges replaced byrubber bands

2( )i jij E

u uE

Energy: Equilibrium:( )

1i j

j N ii

u ud

Rubber band representation Tutte (1963)

Discrete harmonic and analytic functions

Rubber band representation

G 3-connected planar

rubber band embedding is planar

Tutte

(Easily) polynomial time computable

Lifts to Steinitz representation if

outer face is a triangle

Maxwell-CremonaDemo!

Every planar graph can be represented by touching circles

Coin representation Koebe (1936)

Discrete version of the Riemann Mapping Theorem

# ≤ #faces (Euler)

= #edges - #nodes + 2

≤ 2 #nodes - 4

< 2 #nodes

Andre’ev

Every 3-connected planar graph

is the skeleton of a convex polytope

such that every edge

touches the unit sphere

Coin representation Polyhedral version

Coin representation From polyhedra to circles

horizon

Coin representation From polyhedra to representation of the dual

G: connected graph

Roughly: multiplicity of second largest eigenvalue

of adjacency matrix

But: non-degeneracy condition on weightings

Largest has multiplicity 1.

But: maximize over weighting the edges and diagonal entries

The Colin de Verdière number

M=(Mij): symmetric VxV matrix

M has =1 negative eigenvalue

Mii arbitrary

Strong Arnold Property

( ) max corank ( )G M

( )ijX X symmetric,

X=00ijX ij E i j for and

0,MX

normalization

Mij

<0, if ijE

0, if ,ij E i j

The Colin de Verdière number Formal definition

Dimension of solutions of certain PDE’s

μ(G) is minor monotone

deleting and contracting edges

μ≤k is polynomial timedecidable for fixed k

for μ>2, μ(G) is invariant under subdivision

The Colin de Verdière number Basic Properties

μ(G)≤1 G is a path

μ(G)≤2 G is outerplanar

…

μ(G)≥n-4 complement G is planar_

~

Kotlov-L-Vempala

μ(G)≤4 G is linklessly embedable in 3-space

μ(G)≤3 G is a planar

Colin de Verdière, using pde’sVan der Holst, elementary proof

The Colin de Verdière number Special values

1

2

11 21 1

12 22 2

12

2

22 2

1

...

...

. .

...

.

:

n

x x x

x x x

x

x

x x

u

u

u

x x

basis of nullspace of M

Representation of G in

The Colin de Verdière number Nullspace representation

connected

like convex polytopes?

or…

Discrete version of Courant’s Nodal Theorem

The Colin de Verdière number Van der Holst’s Lemma

G 3-connected

planar nullspace representation gives

planar embedding in 2

The vectors can be rescaled so that we get a convex polytope.

The Colin de Verdière number Steinitz representation

Colin de Verdière matrix M

Steinitz representationP

u vq

p

- ( )ijMp q u v

The Colin de Verdière number Steinitz representation

G 4-connected

linkless embed.

nullspace representation gives

linkless embedding in 3

?

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

The Colin de Verdière number Nullspace representation III