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Mathematics Combinatorics Graph Theory Topological Graph Theory David Craft

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A graph is a set of vertices (or points) together with a set of vertex-pairs called edges.

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A graph is a set of vertices (or points) together with a set of vertex-pairs called edges. A graph is a set of vertices (or points) together with a set of vertex-pairs called edges. Graph Theory is the study of graphs.

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An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross.

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An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbedding

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An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbeddingAn imbedding

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An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbeddingAn imbedding Topological Graph Theory is the study of imbeddings of graphs in various surfaces or spaces

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Orientable surfaces (without boundary): sphere S 0

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Orientable surfaces (without boundary): sphere S 0 torus S 1 Orientable surfaces (without boundary): sphere S 0 torus S 1

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Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2

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Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n

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Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n The surface S n is said to have genus n

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Some graphs cannot be imbedded in the sphere… ? ? ?

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? ? …but all can be imbedded in in a surface of high enough genus.

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The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n.

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The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n. For G =the answer is n = 1.

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The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n. For G =the answer is n = 1. For G = the answer is n = 3.

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Maps. Graphs on Surfaces We are mainly interested in embeddings of graphs on surfaces: : G ! S. An embedding should be differentiated from immersion.

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