By Herbert S. Wilf

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Loop. 5. Suppose H(0) = 1 and H(n) ≤ 1 + 2 6. If Q(0) = 0 and Q(n) ≤ n + n i=1 1 n n i=1 H(i − 1) (n ≥ 1). How big might H(n) be? Q(i − 1) (n ≥ 1), how big might Q(n) be? 7. (Research problem) Find the asymptotic behavior, for large n, of the probability that a randomly chosen permutation of n letters has a splitter. 3 Recursive graph algorithms Algorithms on graphs are another rich area of applications of recursive thinking. Some of the problems are quite different from the ones that we have so far been studying in that they seem to need exponential amounts of computing time, rather than the polynomial times that were required for sorting problems.

In Fig. 7(a) we show a graph G of 7 vertices and a chosen edge e. The two endpoints of e are v and w. In Fig. 7(b) we show the graph G/{e} that is the result of the construction that we have just described. Fig. 7(a) Fig. 1. Let v and w be two vertices of G such that e = (v, w) ∈ E(G). Then the number of proper K-colorings of G − {e} in which v and w have the same color is equal to the number of all proper colorings of the graph G/{e}. Proof: Suppose G/{e} has a proper K-coloring. Color the vertices of G − {e} itself in K colors as follows.

How many maximal independent sets does G have? 14. Describe the complement of the graph G in exercise 13 above. How many cliques does it have? 15. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, 3} a triangle? 16. Let H be a labeled graph of L vertices. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, . , L} equal to H? 17. Devise an algorithm that will decide if a given graph, of n vertices and m edges, does or does not contain a triangle, in time O(max(n2 , mn)).