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Oliver Knill Harvard University
December 27, 2014
Thanks to the Harvard College Research Program HCRP for supporting work with
Jenny Nitishinskaya
from June 10-August 7, 2014
work which initiated this research on graph coloring.
COLORING GRAPHS USING TOPOLOGY Oliver Knill Harvard University - - PowerPoint PPT Presentation
COLORING GRAPHS USING TOPOLOGY Oliver Knill Harvard University - - PowerPoint PPT Presentation
COLORING GRAPHS USING TOPOLOGY Oliver Knill Harvard University December 27, 2014 http://arxiv.org/abs/1412.6985 Thanks to the Harvard College Research Program HCRP for supporting work with Jenny Nitishinskaya from June 10-August 7, 2014
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2 DIM SPHERES
S2
= { G | S(x) is cyclic C with n(x)>3 and χ(G)=2
n
every unit sphere
}
χ(G)=v-e+f
Euler characteristic in two dimensions
``spheres have circular unit spheres”
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POSITIVE CURVATURE
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AN OTHER SPHERE?
``not all triangulations are spheres!”
this graph is not maximal planar.
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WHITNEY GRAPHS
W
= {G | G is 4-connected and
maximal planar }
``stay connected if 1,2 or 3 vertices are knocked out” not max
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WHITNEY THEOREM
W
Every Gε
is Hamiltonian
1 2 3 4 5 6
``Hamiltonian connection”
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COMBINATION
``twin octahedron is 4 -disconnected” ``torus is non-planar. with χ=0”
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SPHERE LEMMA
W = S2
``Whitney graphs are spheres”
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4 COLOR THEOREM
P = planar graphs
C =
4
4 -colorable graphs
P ⊂ C
4
``only computer proof so far”
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from Tietze, 1949
MAP COLORING
``Switzerland”
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Schaffhausen Appenzell Geneva
GRAPH
``almost a disk”
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ON SPHERE
``on the globe”
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REFORMULATION
P ⊂ C
4
S ⊂ C
4 2
⇔
``Need only to color spheres”
P ⊂ C
4
C C S ⊂ C
4
C C
2
S S
⇔
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VERTEX DEGREE
``loop size in dual graph”
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KEMPE-HEAWOOD
S ∩
3
E
=
C E = Eulerian graphs
2
S ∩
2
= {all vertex degrees are even} ``Euler”
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CONTRACTIBLE
``inductive setup”
G contractible if there is x such that S(x) and G-B(x) are contractible
∅
contractible graph is
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G
GEOMETRIC GRAPH
S
- 1
``inductive definitions”
B
- 1G
- 1
= = ={ ∅ }
Gd = { G | all S(x) ℇ S
d-1 }
Bd
{
= G ℇ |
d
Sd = { G ℇ | G-{x} ℇ } Gd B
d
δG
ℇ S
d-1
contractible
G
}
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EXAMPLES
S
``does the right thing”
G 0
B S
1
B
1
G1
S
2
B
2
G2
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3 DIM SPHERES
S
3 = {Unit spheres in
``dimension + homotopy”
S
2
+ punching a hole makes
graph contractible }
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DIMENSION
``inductive dimension” dim(G) = 1+E[dim(S(x))]
E[X] = average over all vertices, with counting measure
dim(∅) = -1
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EDGE DEGREE
``loop size in dual graph”
- dd degree is
- bstruction
to color minimally
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CONSERVATION LAW
``is twice edge size on S(x) if x is interior”
∑ deg(e)
x in e
is even
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``red are
- dd degrees”
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MINIMAL COLORING
S ∩
4
E
=
C E = Euler 3D graphs
3
S ∩
3
= {all edge degrees are even}
3 3
``from 1970ies”
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MOTIVATION
S
Every G ε boundary of a H
1
ε B
2∩ E
``silly as trivial, but it shows main idea” is the
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``cut until Eulerian”
PROOF
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CONJECTURE
S
Every G ε is boundary of H
2
ε B
3∩ E 3
``we would see why the 4 color theorem is true”
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REFINEMENTS
``cut an edge”
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embed refine color
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``refine!”
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LETS TRY IT!
``does it work?”
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DECAHEDRON
color that
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0-cobordant
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cut
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cut again
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Eulerian 3D
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colored!
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“by tetrahedra!”
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COBORDISM
``Poincare” ‘
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OCTA-ICOSA
``Cobordism between spheres”
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SELF-COBORDISM
``Sandwich dual graph”
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SELFCOBORDISM
``X=X in cobordism group”
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SELF-COBORDISM
``crystal ”
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EXAMPLE COLORING
``yes it does”
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SIMULATED ANNEALING
``does it always work?”
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CUTTING STEP
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BEYOND SPHERES
G ∩ C
3
``for c=5: Fisk theory” not empty
2,g,o
G ∩ C
4
not empty
2,g,o
G ∩ C
5
not empty
2,g,o
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FISK GRAPH
``Dehn twist”
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JENNY’S GRAPH
``a projective plane of chromatic number 5!”
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3COLORABLE
``A projective plane with minimal color”
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HIGHER GENUS
``glueing game”
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5 COLOR CONJECTURE
``motivated from Stromquist-Albertson type question for tori”
G ⊂ C
5 2
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”Klein bottle” ”Torus”
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D+2 COLOR CONJECTURE
``higher dimensional analogue of 4 color problem”
S ⊂ C
d+2 d
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WHY?
”Embed sphere G in d+1 dimensional minimally colorable sphere H.”
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16 CELL
S
3
in color is 4th dim
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16 CELL
S
3
in colored
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600 CELL
S
3
in color is 4th dim
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CAPPED CUBE
S
3
in 4 colored
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