The Point Partition Numbers of Closed 2-manifolds

Let M be a closed 2-manifold. The chromatic number of M is defined to be the maximum chromatic number of all graphs which can be imbedded in M. The famous Four Colour Conjecture states that the chromatic number of the sphere is four. One of the oddities of mathematics is that the chromatic number of the familiar sphere is still unknown, although the chromatic number of every other closed 2-manifold, whether oiientable or non-orientable, is known. In 1959, Ringel published in [6] his proof that the chromatic number of the closed non-orientable 2-manifold of genus y (y > 0) is [|(7+V(l+24y))] if y # 2, and the chromatic number of the Klein bottle (y = 2) is six. More recently, Ringel and Youngs in [7], 1968, announced their solution to the long-standing Heawood Map-Colouring Conjecture: The chromatic number of the closed orientable 2-manifold of genus y (y > 0) is [i(7 +V(l + 48y))]. A colouring number for graphs closely related to the chromatic number is the point-arboricity (see [2]). The point-arboricity of the closed 2-manifold M is defined to be the maximum point-arboricity of all graphs which can be imbedded in M. In 1969, Kronk showed in [4] that the point-arboricity of the closed orientable 2-manifold of genus y (y > 0) is [i(9+VO+48y))]. Chartrand and Kronk, also in 1969, proved in [1] that the point-arboricity of the sphere is three. The similarity of the three results mentioned above suggested the generalization treated in this paper.