1. Bipartite Kneser graphs are Hamiltonian
Torsten Mütze
joint work with Pascal Su (ETH Zurich)

2. Hamilton cycles
Hamilton cycle = cycle that visits every vertex exactly once

3. Hamilton cycles Problem: Given a graph, does it have a Hamilton cycle?
fundamental problem with many applications (special case of travelling salesman problem)
computational point of view:
no efficient algorithm known (NP-complete [Karp 72]), i.e. brute-force approach essentially best possible
what about particular families of graphs?

4. Kneser graphs Integer parameters and
Vertices = all -element subsets of
Edges =
Examples
iff
Petersen graph
Complete graph
{1,2}
{3,5}
{3,4}
{4,5}
{2,3}
{1,5}
{1,4}
{2,5}
{1,3}
{2,4}
{1}
{4}
{2}
{3}

5. Bipartite Kneser graphs
Integer parameters and
Vertices = all -element and element subsets of
iff
Edges =
Examples 1
{1}
{2}
{3}
{4}
{2,3,4}
{1,3,4}
{1,2,4}
{1,2,3}
{1}
{2}
{3}
{1,2}
{1,3}
{2,3}

6. The middle levels graph
100
010
001
101
011
000
110
111 1 2 3
{1}
{2}
{3}
{1,2}
{1,3}
{2,3}

7. The middle levels graph
100
010
001
101
011
000
110
111 1 2 3
11...1
00...0

8. The middle levels graph
100
010
001
101
011
000
110
111 1 2 3
{1}
{2}
{3}
{1,2}
{1,3}
{2,3}

9. Bipartite Kneser graphs
Integer parameters and
Vertices = all -element and element subsets of
Edges =
Properties
iff
bipartite, connected
number of vertices:
degree:
automorphisms = renaming elements + taking complement
vertex-transitive

10. Is hamiltonian? Conjecture: For all and the graph has Hamilton cycle.
raised by [Simpson 91], and Roth (see [Gould 91], [Hurlbert 94])
Motivation:
Conjecture [Lovász 70]: Every connected vertex-transitive graph has a Hamilton cycle (apart from five exceptions).

11. Is hamiltonian? Conjecture: For all and the graph has Hamilton cycle.
middle levels conjecture
(‚revolving door conjecture‘) 15 x 14 13 12 11 10 9 8 7 6 5 4 3 1 2
first mentioned in [Havel 83], [Buck, Wiedemann 84]
also (mis)attributed to Dejter, Erdős, Trotter [Kierstead, Trotter 88]
exercise (!!!) in [Knuth 05]

12. Is hamiltonian?
attracted considerable attention: [Savage 93] [Felsner, Trotter 95] [Shields, Winkler 95] [Johnson 04] [Moews, Reid 99] [Shimada, Amano 11] [Kierstead, Trotter 88] [Duffus, Sands, Woodrow 88] [Dejter, Cordova, Quintana 88] [Duffus, Kierstead, Snevily 94] [Horák, Kaiser, Rosenfeld, Ryjácek 05] [Gregor, Škrekovski 10] …
finally solved in [M. 15+]
middle levels conjecture 15 x 14 13 12 11 10 9 8 7 6 5 4 3 1 2

13. Is hamiltonian? ??? Known results: has a Hamilton cycle if
[Shields, Savage 94]
[Chen 03] (following earlier work by [Simpson 94], [Hurlbert 94], [Chen 00])
??? 15 x 14 13 12 11 10 9 8 7 6 5 4 3 1 2

14. Our results Theorem: For all and the graph has Hamilton cycle. Remark:
simple induction proof, assuming the validity of the middle levels conjecture

15. Key lemma Lemma: For all and there is a cycle and a set of
vertex-disjoint monotone paths
in such that:

16. Lemma  Theorem Lemma: For all and there is a cycle and a set of
vertex-disjoint monotone paths
in such that: z6 z1 z2 z3 z5 z4 y6 y1 y2 y3 y4 y5 x1 x2 x3 x4 x5 x6

17. Key lemma Lemma: For all and there is a cycle and a set of
vertex-disjoint monotone paths
in such that:
00…011…1
00…011…10

18. Proof of lemma proof by induction ‚manual‘ middle levels construction
conjecture
proof by induction
‚manual‘
construction 15 x 14 13 12 11 10 9 8 7 6 5 4 3 1 2

19. Key lemma Lemma: For all and there is a cycle and a set of
vertex-disjoint monotone paths
in such that:
00…011…1
00…011…10

20. Induction step

21. Induction step

22. Induction step

23. Induction step Y X

24. Induction step

25. Induction step

26. Thank you!