1. Prove that it is impossible to have a group of nine people at a party...

1. Prove that it is impossible to have a group of nine people at a party such that each one knows exactly five others in the group.
2. Let G be a graph with n vertices, t of which have degree k and the others have degree k+1. Prove that t = (k+1)n - 2e, where e is the number of edges in G.
3. Let G be a k-regular graph, where k is an odd number. Prove that the number of edges in G is a multiple of k.
4. Let G be a graph with n vertices and exactly n-1 edges. Prove that G has either a vertex of degree 1 or an isolated vertex.
5. Show that the k-cube has 2^k vertices and k2^(k-1) edges and is bipartite.
6. Prove that if G is a simple graph with at least two vertices, then G has two or more vertices of the same degree.

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Colby Messinger answered 2 years ago

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