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

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

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

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

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

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

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

10. 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 Rubber band representation Tutte Demo!

11. 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. 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. 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. <0, if ijE Mij 0, if symmetric,  X=0 The Colin de Verdière number Formal definition M=(Mij): symmetric VxV matrix Miiarbitrary normalization M has =1 negative eigenvalue Dimension of solutions of certain PDE’s Strong Arnold Property

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

19. 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 The Colin de Verdière number Special values μ(G)≤1G is a path μ(G)≤2G is outerplanar

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

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

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

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

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