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10. Draw all of the connected, unlabeled graphs of 4 vertices. 11. Let G be a bipartite graph that has q connected components. Show that there are exactly 2q ways to properly color the vertices of G in 2 colors. 12. Find a graph G of n vertices, other than the complete graph, whose chromatic number is equal to 1 plus the maximum degree of any vertex of G. 6 Graphs 13. Let n be a multiple of 3. Consider a labeled graph G that consists of n/3 connected components, each of them a K3 . How many maximal independent sets does G have?

To do this we simply have to look at the input data. In the worst case we might look at all of the input data, all Θ(n2 ) bits of it. Then, if G actually has some edges, the additional labor needed to process G consists of two recursive calls on smaller graphs and one computation of the larger of two numbers. If F (G) denotes the total amount of computational labor that we do in order to find maxset1(G), then we see that F (G) ≤ cn2 + F (G − {v∗ }) + F (G − {v∗ } − N bhd(v∗ )). 1) over all graphs G of n vertices.

8. True or false: a Hamilton circuit is an induced cycle in a graph. 9. Which graph of n vertices has the largest number of independent sets? How many does it have? 10. Draw all of the connected, unlabeled graphs of 4 vertices. 11. Let G be a bipartite graph that has q connected components. Show that there are exactly 2q ways to properly color the vertices of G in 2 colors. 12. Find a graph G of n vertices, other than the complete graph, whose chromatic number is equal to 1 plus the maximum degree of any vertex of G.

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