Transcript Proof of the middle levels conjecture

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?
Bipartite Kneser graphs
• Integer parameters
and
• Vertices = all -element and
subsets of
• Edges =
-element
iff
• Examples
{2,3,4} {1,3,4} {1,2,4} {1,2,3}
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{1}
{2}
{3}
{4}
{1,2}
{1,3}
{2,3}
{1}
{2}
{3}
The middle levels graph
111
3
2
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011
1
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001
0
{1,2}
{1,3}
{2,3}
{1}
{2}
{3}
000
The middle levels graph
11...1
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001
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00...0
Bipartite Kneser graphs
• Integer parameters
and
• Vertices = all -element and
subsets of
• Edges =
-element
iff
• Properties
• bipartite, connected
• number of vertices:
• degree:
• automorphisms = renaming elements + taking complement
• vertex-transitive
Is
Conjecture:
For all
and
hamiltonian?
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
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has Hamilton cycle.
middle levels conjecture
(‚revolving door conjecture‘)
• first mentioned in
[Havel 83], [Buck, Wiedemann 84]
• also (mis)attributed to Dejter,
Erdős, Trotter [Kierstead, Trotter 88]
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• exercise (!!!) in [Knuth 05]
Is
hamiltonian?
• attracted considerable
attention:
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[Savage 93]
[Felsner, Trotter 95]
[Shields, Winkler 95]
[Johnson 04]
[Moews, Reid 99]
[Shimada, Amano 11]
middle
levels
conjecture
[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+]
Is
hamiltonian?
Known results:
has a Hamilton cycle if
•
[Shields, Savage 94]
•
[Chen 03]
(following earlier work by [Simpson 94], [Hurlbert 94], [Chen 00])
???
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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 x 3 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‘
construction
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middle levels
conjecture
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
Induction step
Induction step
Y
X
Induction step
Thank you!