A tighter bounding interval for the 1-chromatic number of a surface Original Research Article

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 over by no more than one other edge. In the previous paper the author has proved that F(S) − 34 ⩽ χ1(S), where F(S)=⌊12(9 + √(81−32E(S)))⌋ is Ringelʹs upper bound for χ1(S) and E(S) is the Euler characteristic of S. In this paper it is shown that F(S) − 10 ⩽ χ1(S).