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It is easy to imagine that some of those lists are very long, and that the replacement of Θ(n2 ) by an average of O(n log n) comparisons is very welcome. An insurance company that wants to alphabetize its list of 5,000,000 policyholders will gratefully notice the difference between n2 = 25, 000, 000, 000, 000 comparisons and n log n = 77, 124, 740 comparisons. If we choose as our unit of complexity the number of swaps of position, then the running time may depend strongly on the input array. ).

Hence, suppose it is true for all graphs of V vertices and fewer than E edges, and let G have V vertices and E edges. 4) implies that P (K; G) is a polynomial of the required degree V because G − {e} has fewer edges than G does, so its chromatic polynomial is a polynomial of degree V . G/{e} has fewer vertices than G has, and so P (K; G/{e}) is a polynomial of lower degree. The claim is proved, by induction. 4) gives a recursive algorithm for computing the chromatic polynomial of a graph G, since the two graphs that appear on the right are both ‘smaller’ than G, one in the sense that it has fewer edges than G has, and the other in that it has fewer vertices.

Here are some special families of graphs that occur so often that they rate special names. The complete graph Kn is the graph of n vertices in which every possible one of the n2 edges is actually present. Thus K2 is a single edge, K3 looks like a triangle, etc. , has no edges at all. The complete bipartite graph Km,n has a set S of m vertices and a set T of n vertices. Its edge set is E(Km,n ) = S × T . It has |E(Km,n )| = mn edges. The n-cycle, Cn , is a graph of n vertices that are connected to form a single cycle.