In this note a graph G is a finite 1-complex, and an imbedding of G in an orientable 2-manifold M is a geometric realization of G in M.The letter G will also be used to designate the set in M which is the realization of G. Manifolds will always be orientable 2-manifolds, and y(M) will stand for the genus of M. Given a graph G the genus y(G) of G is the smallest number y(M), for M in the collection of manifolds in which G can be imbedded.A block of G is a subgraph B of G maximal with respect to the property that removing any single vertex of B does not disconnect 5, (A block with more than two vertices is a "true cyclic element" in Whyburn
In a complete graph every two points are joined by a line (are adjacent). A complete graph with n points is denoted by K n > Let G be a graph with n points considered as a subgraph of K n . The complement G of G is the graph obtained by removing all lines of G fromi£ n . The following problem was stated by Harary [2 ] : What is the least integer n such that every graph G with n points or its complement G is nonplanar? Harary [3] observed that n^ll.It is readily seen that n>8. In this note we shall outline the proof that n = 9, verifying a conjecture of J. L. Self ridge. THEOREM. If G is a graph with nine points, then G or its complement G is nonplanar.Let p(G) be the number of points, q(G) the number of lines, and k(G) the number of components of graph G. Let K m , n be a graph with m+n points, m points of one color and n points of another, in which two points are adjacent if and only if their colors are different. Kuratowski [5 ] proved the classic theorem that a graph is nonplanar if and only if it contains a subgraph homeomorphic to KB or Kz,z.
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