Noncrossing Hamiltonian Paths

1. Noncrossing Hamiltonian Paths Jakub Černý, Zdeněk Dvořák, Vít Jelínek, Jan Kára

2. Definitions • Geometric Graph: vertices: points in the plane in general position (no three in a line) edges: segments • Noncrossing Hamiltonian Path: a path which contains all the vertices and does not intersect itself

3. An Observation

4. An Observation

5. There is a positive constant c such that The Main Problem • f(n) = max {k; any geometric graph on n vertices whose complement has at most k edges has a noncrossing Hamiltonian path} • The problem (Perles, 2002): determine the value of f(n)

6. Observation: • Result: A Special Case • B(n) = max {k; when we remove the edges of any clique of size k from any complete geometric graph on n vertices, the remaining graph has a noncrossing Hamiltonian path}

10. L S

11. The Upper Bound

12. Vertices in Convex Position

13. Complements of a Star • The missing edges form a star:

14. Complements of a Matching • The missing edges form a matching: a noncrossing Hamiltonian path always exists

15. Open Problems • f(n) = ? • g(n) = max {k; every complete geometric graph on n vertices contains k disjoint noncrossing Hamiltonian paths} • h(n) = max {k; for each complete geometric graph G there is a constant m and a multiset F of mk noncrossing Hamiltonian paths in G such that every edge is contained in at most m of them}