By C. Ward Henson, José Iovino, Alexander S. Kechris, Edward Odell, Catherine Finet, Christian Michaux

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Hopcroft and R. E. e. in O(V ) time for a graph of V vertices. Although every planar graph can be properly colored in four colors, there are still all of those other graphs that are not planar to deal with. For any one of those graphs we can ask, if a positive integer K is given, whether or not its vertices can be K-colored properly. As if that question weren’t hard enough, we might ask for even more detail, namely about the number of ways of properly coloring the vertices of a graph. For instance, if we have K colors to work with, suppose G is the empty graph K n, that is, the graph of n vertices that has no edges at all.

3 Recursive graph algorithms Fig. 5(a) Fig. 5(b) of this construction, we show in Fig. 5(a) a map of a distant planet, and in Fig. 5(b) the graph that results from the construction that we have just described. By a ‘planar graph’ we mean a graph G that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph.

Finally we call chrompoly on the graph G/{e}. Let F (V, E) denote the maximum cost of calling chrompoly on any graph of at most V vertices and at most E edges. 5) together with F (V, 0) = 0. If we put, successively, E = 1, 2, 3, we find that F (V, 1) ≤ c, F (V, 2) ≤ 4c, and F (V, 3) ≤ 11c. 6) then we will have such a solution. 5), we find that E f(E) = 2E j2−j j=0 ∼ 2E+1. 3. To summarize the developments so far, then, we have found out that the chromatic polynomial of a graph can be computed recursively by an algorithm whose cost is O(2E ) for graphs of E edges.