By Jean Pierre Serre

Jean-Pierre Serre is Professor on the Collège de France. He has written a couple of books, together with "Algebraic teams and sophistication Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", accrued Papers (3 volumes), and "Trees" released by means of Springer-Verlag.

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Redraw the graph with the hamiltonian cycle on the exterior. 4. 31 Example A decomposition of the complete graph into triangles. 26 shows that K7 can be decomposed into edge-disjoint copies of K3 (the triangle). The toroidal embedding is not needed to see this, however. 31 explicitly shows seven K3 s that edge-partition K7 . Another line of investigation opens: When can one edge-partition a graph G into copies of a graph H? Combinatorial design theory concentrates on cases when G is a complete graph.

He also wrote that he thought that a solution for 6 is “improbable,” for 7 is “very likely” (because 7 is prime) and that he did not see why one should not be discovered for 9. He did not comment on order 10. A few years earlier, Kirkman [1301] had shown how to construct affine planes of prime order, and, hence, essentially by adding points at infinity, finite projective planes of all prime orders (although he did not use the geometric terminology). This paper also showed how to construct certain families of pairwise balanced designs.

He showed how to construct such of order 2n from a solution of order n; twenty-six years later Hadamard [1003] showed that Hadamard matrices give the largest possible determinant for a matrix whose entries are bounded by 1. Hadamard showed that the order of such a matrix had to be 1, 2, or a multiple of 4, and he constructed matrices of orders 12 and 20. In 1898 Scarpis [1846] showed how to construct Hadamard matrices of order 2k · p(p + 1) whenever p is a prime for which a Hadamard matrix of order p + 1 exists.