A lower bound for the one-chromatic number of a surface (Q1328382)

Let \(\chi_ 1(S)\) 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 \(F(S)-34\leq\chi_ 1(S)\), where \(F(S)=\lfloor{1\over 2}(9+\sqrt{81-32E(S)})\rfloor\) is Ringel's upper bound for \(\chi_ 1(S)\) and \(E(S)\) is the Euler characteristic of \(S\).