Torsten Mütze ( based on - PowerPoint Presentation

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Torsten Mütze(based on joint work with Karl Däubel, Sven Jäger, Petr Gregor, Joe Sawada, Manfred Scheucher, and Kaja Wille)

On symmetric chains and Hamilton cycles

The Boolean lattice

consider

all

subsets

of

ordered

by inclusion

4-cube

a

fundamental and widely studied poset

its

size is

called

-

cube

-th

level

:= all

subsets of cardinality

The Boolean lattice

consider

all

subsets

of

ordered

by inclusion

even

odd

middle

level

(s)

a

fundamental and

widely

studied

poset

its

size

is

called

-

cube

-th

level

:= all

subsets

of

cardinality

Chain decompositions

Theorem

[

Sperner

28]

:

The width (=size of a maximum

antichain) of the -cube is

given by the size of its middle level(s) .

Theorem

[

Dilworth 50]:Any poset can be decomposedinto many

chains.

4-cube

chain

decomposition

Symmetric chain decompositions

useful

for

applications

:

symmetric

chains

,

i.e

.,

if

a

chain

starts

at level , then it ends at level .

(SCD)

symmetric

chain

decomposition

known

constructions

of

SCDs

for

the

-cube

due to[De Bruijn, van Ebbenhorst Tengbergen, Kruiswijk 51], [Lewin 72], [Aigner 73], [White and Williamson 77], [Greene, Kleitman 76]all constructions yield the same SCD

not

symmetric

4-cube

Parenthesis matching

useful

for

applications

: symmetric chains, i.e., if a chain starts at level ,

then it ends at

level .

‚parenthesis matching‘

description

by [Greene, Kleitman 76]

1001100100

10011

1

0100

10011101

1

0

100111011

1

0

001100100

known

constructions

of

SCDs

for

the

-cube

due

to

[De

Bruijn

, van Ebbenhorst Tengbergen, Kruiswijk

51], [Lewin 72], [Aigner 73], [White and Williamson 77], [Greene, Kleitman 76]

all

constructions

yield the same

SCD

Edge-disjoint

and orthogonal SCDs

Question

:

Are

there

other

constructions?

4-cube

Definition:

T

wo

SCDs

are

edge-disjoint

,

if

they

do not

share

any

edges

4-cube

Definition:

T

wo

SCDs

are

orthogonal

,

if

any

two

chains

intersect

in at

most

one

element

,

except

the

two

longest

chains

that

may

only

intersect

in and

Observe

:

orthogonal



edge-disjoint

even

Edge-disjoint

and orthogonal

SCDs

Question

:

How

many

pairwise

edge-disjoint

/orthogonal

SCDs

can

we

hope for?

is

an

upper

bound

:

Conjecture

[Shearer,

Kleitman

79]

: The -cube has pairwise orthogonal SCDs.

every

SCD

uses exactly

one of those edges

Theorem

[Shearer,

Kleitman 79]: The standard

construction and its complements

are two

orthogonal SCDs.

Theorem

[Spink

17]: The -cube has three

pairwise orthogonal SCDs for

.

Our results

Theorem 1:

The

-cube

has

four

pw. orthogonal SCDs for .

Theorem 2:

The

-cube has five pw. edge-disjoint SCDs for .

Proof

of Theorem 2:

Proof of Theorem 1:

Product

lemma

:

If the -cube

and -cube have edge-disjoint SCDs each, then the -cube has edge-disjoint SCDs.

find

five

edge-disjoint

SCDs

for

dimensions

and

Fact

:

If

and

are coprime, then every is a non-negative integer multiple of and .similar, but more complicated product lemma due to [Spink 17]

find

four orthogonal SCDs for dimensions and

computer search in the necklace

poset

The

central

levels

problem

-cube

Middle

levels

conjecture

:

The

subgraph

of the

-cube

induced

by

the

middle

two

levels

and has a Hamilton

cycle

.

Central

levels

co

njecture

:

The

subgraph

of the

-cube

induced

by

the

middle

levels

has a Hamilton

cycle

for

any

.

answered

positively

in

[M. 16]

raised

by

[

Savage

93], [

Gregor

,

Škrekovski

10], [

Shen

, Williams

15]

problem

with

a

long

history

The

central

levels

problem

[M. 16]

[Gray 53]

[

El-

Hashash

, Hassan 01], [Locke, Stong 03]

[

Gregor

, Škrekovski 10]

???

-cube

known

results:

Central

levels

co

njecture

:

The

subgraph

of the

-cube

induced

by

the

middle

levels

has a Hamilton

cycle

for

any

.

Theorem 4:The -cube has a cycle factor through the middle levelsfor all and .

Our

results

Theorem

3:

The

-cube

has

a

Hamilton

cycle

through

the middle four

levels ( ) for all .

spanning

collection

of disjoint cycles

The

central

levels

problem

Proof:

consider

two edge-disjoint SCDs

as the dimension is odd, all chains have odd length

, even

after restricting to middle levels

taking

every second edge

yields two edge-disjoint perfect matchings

their

union

is a cycle factor

Theorem

4

:

The

-cube

has

a

cycle

factor

through

the

middle

levels

for

all and .

Open problemsConjecture [Shearer, Kleitman 79]: The -cube has pairwise orthogonal SCDs.

we

conjecture

that

the -cube has pairwise edge-disjoint SCDs.known: fourknown:

fivecentral levels problem: Can the cycles in the factor be

joined to a single Hamilton cycle?

Structure of the cycle factor?

exploit new SCD constructions in other applications (Venn diagrams etc.)

efficient

algorithms to generate those cycles

first open case:

middle six levels

Thank you!