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