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