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By Herbert S. Wilf

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Then we will distinguish two kinds of proper colorings of G−{e}: those in which vertices v and w have the same color and those in which v and w have different colors. Obviously the number of proper colorings of G − {e} in K colors is the sum of the numbers of colorings of each of these two kinds. * Efficient planarity testing, J. Assoc. Comp. Mach. 21 (1974), 549-568. 43 Chapter 2: Recursive Algorithms Consider the proper colorings in which vertices v and w have the same color. We claim that the number of such colorings is equal to the number of all colorings of a certain new graph G/{e}, whose construction we now describe: The vertices of G/{e} consist of the vertices of G other than v or w and one new vertex that we will call ‘vw’ (so G/{e} will have one less vertex than G has).

61803.... We chose the ‘ ’ that appears in the conclusion of the theorem simply by rounding c upwards. What have we learned? 619n) units if the input graph G has n vertices. 3 Recursive graph algorithms algorithm that one might think of for this problem, which is to examine every single subset of the vertices of of G and ask if it is an independent set or not. That algorithm would take Θ(2n ) time units because there are 2n subsets of vertices to look at. 619n by being a little bit cagey about the algorithm.

10) follows. 10) is obviously the relation h(γ) ≤ Fγ , where Fγ is the Fibonacci number. 11) ). This analysis does not, of course, contradict the earlier estimate, but complements it. 62|V (G)|+|E(G)|) . 12) will be the smaller one, whereas if G has more edges, then the second term is the smaller one. ) remains. 3 1. Let G be a cycle of n vertices. What is the size of the largest independent set of vertices in V (G)? 2. Let G be a path of n vertices. What is the size of the largest independent set of vertices in V (G)?

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Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf

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