Pillow Talk - PowerPoint

Document Sample
Pillow Talk - PowerPoint Powered By Docstoc
					       Pillow Talk
Finite Hyperbolic Tilings and
    Independence Number

         Emily Ward
Tilings
Tilings
Planar tiling




Hyperbolic tiling




A polyhedron
        Euler’s Formula
       Euler’s Formula
Euler’s Formula: F – E + V = 2 – 2g

Consider 3 heptagons at a vertex.
Relations:              3V = 2E
                        7F = 2E

Substitutions give:    F = 12(g - 1)

Number of sleeves: 12(g-1) = 3 (g-1)
                      4
Ward Diagrams
Ward Diagrams
       The top of a
       sleeve.




  Connecting diagrams
            Sleeve Graphs (G)
             Sleeve Graphs
Sleeve graph (G): A simplified representation of a pillow where
      An Edge = Sleeve
      A Vertex = Meeting of three sleeves.
    Independence Number
    Independence Number
Choose 2 per sleeve.
That gives:
         2 * number of sleeves
         = 2(3(g-1))
         = 6(g-1)
contributed to the independent set from the sleeves.


For example:
Independence Number (cont)
 Independence Number (cont)




V0:             a(V0) = 4
V1:   a(V1) = 3




V2:   a(V2) = 2




V3:    a(V3) = 2
Independence Number (cont)
Independence Number (cont)
(1)   | i | = <V0, V1, V2, V3> . <4, 3, 2, 2> + 6(g-1)


(2)   <V0, V1, V2, V3> . <1, 1, 1, 1> = 2(g-1)
(3)   <V0, V1, V2, V3> . <0, 1, 2, 3> = 3(g-1)


(4)   | i | = V3 + 11(g-1)
       Kernel
      Maximizing




Graph G               Kernel of G
                          K(G)


 \ Maximum V3 = a(K(G))
   YAY ANSWER!
Independence Number


a(PG) = 11(g-1) + a (K(G))

				
DOCUMENT INFO
Shared By:
Categories:
Stats:
views:28
posted:5/13/2010
language:English
pages:13