Minimal decompositions of complete graphs into subgraphs with embeddability properties

The surfaces which will be considered here are the orientable surfaces Sn obtained from the sphere by adding n handles and the non-orientable surfaces S obtained from the sphere by making n crosscuts. The corresponding characteristics are of course 2 — 2n and 2 — n. The corresponding thicknesses of the graph G will be denoted by tn(G) and t [n] (G). For the plane (or sphere), the inequality (1) has been shown (1) to be equality for "five-sixths" of the complete graphs Kp. The known results are summarized in the following theorem. THEOREM 1. If p 9 9 and p ^ 4 (mod 6), then the planar thickness of the complete graph is