Unformatted text preview: (6) Show that if a simple graph G is isomorphic to its complement G , then G has either 4 k or 4 k + 1 vertices for some natural number k . Find all simple graphs on four and ﬁve vertices that are isomorphic to their complements. (7) The complete bipartite graphs K 1 ,n , known as the star graphs , are trees. Prove that the star graphs are the only complete bipartite graphs which are trees. (8) A graph G is bipartite if there exists nonempty sets X and Y such that V ( G ) = X ∪ Y , X ∩ Y = ∅ and each edge in G has one endvertex in X and one endvertex in Y . Prove that any tree with at least two vertices is a bipartite graph....
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- Spring '11
- Graph Theory, Vertex, vertices, independent set, Bipartite graph, simple graph