A Lower Bound for the One-Chromatic Number of a Surface

Let χ1(S) be the maximum chromatic number for all graphs which can be drawn on a surface S so that each edge is crossed by no more than one other edge. It is proved that F(S) − 34 ≤ χ1(S), where F(S) = ⌊12(9 + [formula]) ⌋ is Ringel′s upper bound for χ1(S) and E(S) is the Euler Characteristic of S.