Pillow Talk

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Pillow Talk
Finite Hyperbolic Tilings and Independence Number
Emily Ward
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Tilings
Tilings
Planar tiling
Hyperbolic tiling
A polyhedron
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Eulers Formula
Eulers Formula
Eulers 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)
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Ward Diagrams
Ward Diagrams
The top of a sleeve.
Connecting diagrams
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Sleeve Graphs (G)
Sleeve Graphs
Sleeve graph (G) A simplified representation of
a pillow where
An Edge Sleeve A Vertex Meeting of three
sleeves.
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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
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Independence Number (cont)
Independence Number (cont)
V0
a(V0) 4
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a(V1) 3
V1
V2
a(V2) 2
V3
a(V3) 2
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Independence Number (cont)
4V0 3V1 2V2 2V3 6(g-1)
Independence Number (cont)
(1) i ltV0, V1, V2, V3gt . lt4, 3, 2, 2gt
6(g-1)
(2) ltV0, V1, V2, V3gt . lt1, 1, 1, 1gt 2(g-1)
(3) ltV0, V1, V2, V3gt . lt0, 1, 2, 3gt 3(g-1)
(4) i V3 11(g-1)
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Kernel
Maximizing
Graph G
Kernel of G K(G)
\ Maximum V3 a(K(G))
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YAY ANSWER!
Independence Number
a(PG) 11(g-1) a (K(G))
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