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

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

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?

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}

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}

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

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

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

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

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).

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]

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

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

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

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

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

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

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

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

Induction step

Induction step

Induction step

Induction step Y X

Induction step

Induction step

Thank you!