Embedded graphs. Sasha Patotski. Cornell University November 24, 2014
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1 Embedded graphs Sasha Patotski Cornell University November 24, 2014 Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
2 Exercise Embed K 6 and K 7 into a torus. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
3 2-cell embeddings of graphs Definition A 2-cell embedding of a graph G into a surface Σ is such an embedding, so that after you remove the graph, the resulting surface will be a disjoint union of discs. Theorem Suppose graph G is 2-cell embedded into a sphere with g handles. Then V E + F = 2 2g, where V is the number of vertices, E is the number of edges and F is the number of faces. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
4 Theorem For any graph G there exists a number g so that G can be drawn on a sphere with g handles. Definition Define γ(g) minimal number g, so that G can be embedded into a sphere of genus g. So γ(k 5 ) = γ(k 3,3 ) = 1, γ(k 8 ) 2, γ(tree)? Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
5 Theorem We have γ(k n ) (n 3)(n 4) 12. More generally, for polyhedral graphs, γ(g) 1 V 2 + E 6. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
6 Theorem We have γ(k n ) (n 3)(n 4) 12. More generally, for polyhedral graphs, γ(g) 1 V 2 + E 6. Follows from Euler formula and F 2 3 E. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
7 Theorem We have γ(k n ) (n 3)(n 4) 12. More generally, for polyhedral graphs, γ(g) 1 V 2 + E 6. Follows from Euler formula and F 2 3 E. Corollary: γ(k 8 ) 2. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
8 Euler characteristic Theorem Suppose graph G is 2-cell embedded into a sphere with g handles. Then V E + F = 2 2g, where V is the number of vertices, E is the number of edges and F is the number of faces. Notice: For a given g, any 2-cell decomposition of sphere with g handles has the same number V E + F. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
9 Euler characteristic Theorem Suppose graph G is 2-cell embedded into a sphere with g handles. Then V E + F = 2 2g, where V is the number of vertices, E is the number of edges and F is the number of faces. Notice: For a given g, any 2-cell decomposition of sphere with g handles has the same number V E + F. Notice: 2-cell decomposition makes sense for any surface! Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
10 Euler characteristic Theorem Suppose graph G is 2-cell embedded into a sphere with g handles. Then V E + F = 2 2g, where V is the number of vertices, E is the number of edges and F is the number of faces. Notice: For a given g, any 2-cell decomposition of sphere with g handles has the same number V E + F. Notice: 2-cell decomposition makes sense for any surface! Definition The number V E + F is called Euler characteristic. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
11 Euler characteristic Theorem Suppose graph G is 2-cell embedded into a sphere with g handles. Then V E + F = 2 2g, where V is the number of vertices, E is the number of edges and F is the number of faces. Notice: For a given g, any 2-cell decomposition of sphere with g handles has the same number V E + F. Notice: 2-cell decomposition makes sense for any surface! Definition The number V E + F is called Euler characteristic. So far we know that it is correctly defined for spheres with handles. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
12 Theorem The Euler characteristic is correctly defined for any surface. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
13 Theorem The Euler characteristic is correctly defined for any surface. Any polygon can be triangulated. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
14 Theorem The Euler characteristic is correctly defined for any surface. Any polygon can be triangulated. Euler characteristic stays the same. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
15 Theorem The Euler characteristic is correctly defined for any surface. Any polygon can be triangulated. Euler characteristic stays the same. Euler characteristic is invariant under barycentric subdivision (refinement). Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
16 Theorem The Euler characteristic is correctly defined for any surface. Any polygon can be triangulated. Euler characteristic stays the same. Euler characteristic is invariant under barycentric subdivision (refinement). Euler characteristic is invariant under coarsening. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
17 Coarsening Definition A map of a surface S is a partition of S into properly attached polygons. A coarsening of a triangulation T of S is a map of S in which each polygon is the union of 2-simplices from T. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
18 Coarsening Definition A map of a surface S is a partition of S into properly attached polygons. A coarsening of a triangulation T of S is a map of S in which each polygon is the union of 2-simplices from T. Euler characteristic is invariant under coarsening. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
19 Coarsening Definition A map of a surface S is a partition of S into properly attached polygons. A coarsening of a triangulation T of S is a map of S in which each polygon is the union of 2-simplices from T. Euler characteristic is invariant under coarsening. Idea: have two triangulations T 1 and T 2. We want to find a map M, which is coarsening of some refinement of T 1 and is approximating T 2. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
20 Coarsening Definition A map of a surface S is a partition of S into properly attached polygons. A coarsening of a triangulation T of S is a map of S in which each polygon is the union of 2-simplices from T. Euler characteristic is invariant under coarsening. Idea: have two triangulations T 1 and T 2. We want to find a map M, which is coarsening of some refinement of T 1 and is approximating T 2. This would finish the proof. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
21 Coarsening Definition A map of a surface S is a partition of S into properly attached polygons. A coarsening of a triangulation T of S is a map of S in which each polygon is the union of 2-simplices from T. Euler characteristic is invariant under coarsening. Idea: have two triangulations T 1 and T 2. We want to find a map M, which is coarsening of some refinement of T 1 and is approximating T 2. This would finish the proof. Exercise: compute Euler characteristic of RP 2, K 2, T 2, S 2. Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
22 Attaching a Möbius strip Attaching a handle: Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
23 Attaching a Möbius strip Attaching a handle: Attaching a Möbius band: Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
24 Lemma about attaching Lemma Sphere with one handle and one Möbius band is homeomorphic to a sphere with 3 Möbius bands. Proof: Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
25 Question:How does χ(σ) change when attaching a handle? When attaching a Möbius strip? Sasha Patotski (Cornell University) Embedded graphs November 24, / 11
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