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

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

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

Euler and graph theory The Polyhedron theorem

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 ...

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

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

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.

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:

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!

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

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

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

Coin representation From polyhedra to circles horizon

Coin representation From polyhedra to representation of the dual

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

 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

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

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

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

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?

 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.

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

?  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 