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Graphs and polyhedra:
From Euler to
many branches of mathematics
László Lovász
Eötvös University, Budapest
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Euler and graph theory The Königsberg bridges
Eulerian graphsChinese Postman Problem
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Euler and graph theory The Knight’s Tour
Hamilton cyclesTraveling Salesman Problem
P vs. NP-complete
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Euler and graph theory The Polyhedron theorem
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Euler and graph theory The Polyhedron theorem
#vertices - #edges + #faces = 2
algebraic topology (Euler characteristic)combinatorics of polyhedraMöbius function...
Polyhedra have combinatorial structure!
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Convex polyhedra and planar graphs
3-connected planar graph
For every planar graph, #edges ≤ 3 #nodes - 6
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Planar graphs: straight line representation
Every planar graph can be drawnin the plane with straight edges
Fáry-Wagner
planar graph
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Steinitz 1922Every 3-connected planar graph
is the skeleton of a convex 3-polytope.
3-connected planar graph
Planar graphs and convex polyhedra
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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
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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!
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Every planar graph can be represented by touching circles
Coin representation Koebe (1936)
Discrete version of the Riemann Mapping Theorem
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# ≤ #faces (Euler)
= #edges - #nodes + 2
≤ 2 #nodes - 4
< 2 #nodes
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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
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Coin representation From polyhedra to circles
horizon
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Coin representation From polyhedra to representation of the dual
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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
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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
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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
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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
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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
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connected
like convex polytopes?
or…
Discrete version of Courant’s Nodal Theorem
The Colin de Verdière number Van der Holst’s Lemma
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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
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Colin de Verdière matrix M
Steinitz representationP
u vq
p
- ( )ijMp q u v
The Colin de Verdière number Steinitz representation
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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