Discrete Mathematics With Graph Theory 3rd Edition Free Pdf [VERIFIED]
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The remainder of the proof will be in the language of graph theory. We will say that a node is a friend of a node if the node is not on the outside face. A node is adjacent to a node if the node is incident to an edge that connects the two nodes. For a face, the neighbors are the nodes on the boundary of the face, and they are called boundary nodes. To apply Sperner's lemma, we need to know how we can pick nodes to color. Let's pick a node of each color that has no friends. We need to find the smallest number of friends a single color needs to have.
First of all, if we have an odd number of triangles then the triangulation graph will have an odd number of vertices. To see this, consider the triangles corresponding to the vertices with odd degree. Since the degree of each vertex has an odd number of neighbors with the same color, one vertex of each degree has an odd number of neighbors with the same color as it. The reason the triangulation graph cannot have an even number of vertices is that the outside face has an odd number of neighbors of each color. Hence, the three other minimal triangles must have an odd number of vertices. An application of the Handshaking lemma now finishes the proof. But let's do one more thing: we want to show that the triangles in a triangulation can be arranged in a way that keeps a minimum number of the edges of the original triangle.
The proof of Sperner's lemma and its generalization to higher dimensions is closely related to the proof of Hall's marriage theorem. This is because the proofs of Hall's marriage theorem and Sperner's lemma use an intermediate result, the Handshaking lemma . We'll see later that this relationship is one of the reasons that many proofs in graph theory are self-contained. The Euler characteristic is yet another application of the Handshaking lemma.
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