On the Edge Achromatic Numbers of Complete Graphs

Abstract

Suppose we wish to color the edges of the complete graph K n with as many colors as possible so that (1) no two edges with a common node get the same color, and (2) for any two colors c 1 and c 2 , there are two edges with a common node, one colored c 1 and the other colored c 2 . What is the maximum number A ( n ) of colors possible in such a coloring? Coloring problems are notoriously hard and this problem is no exception. In fact, a remarkable theorem of André Bouchet implies that an exact determination of A ( n ) for all odd n would yield as a corollary all odd orders for which projective planes exist. Thus such a determination is clearly beyond the hopes of this study. The goals here are more modest: (1) to give a careful study of the best available upper bound on A ( n ), (2) to add to the constructions which give reasonable lower bounds for A ( n ), and (3) to contribute a few more values of n for which A ( n ) is known exactly.