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There will be a vertex of G corresponding to each country on the map. Two of these vertices will be connected by an edge of the graph G if the two countries that they correspond to have a common stretch of border (we keep saying ‘stretch of border’ to emphasize that if the two countries have just a single point in common they are allowed to have the same color). As an illustration * Every planar map is four colorable, Bull. Amer. Math. , 82 (1976), 711-712. 43 Fig. 5(a) Fig. 5(b) of this construction, we show in Fig.

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. Then G has quite a large number of proper colorings, K n of them, to be exact.

Let S be one of the equivalence classes of vertices of G under this relation. The subgraph of G that S induces is called a connected component of the graph G. A graph is connected if and only if it has exactly one connected component. , one in which vk = v1 . A cycle is a circuit if v1 is the only repeated vertex in it. We may say that a circuit is a simple cycle. We speak of Hamiltonian and Eulerian circuits of G as circuits of G that visit, respectively, every vertex, or every edge, of a graph G.

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