# Pillow Talk - PowerPoint

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```					       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))