The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n
, consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.
Our results are summarized in the following theorem, where square brackets denote “integer part”.
THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1,