The Positive Lightness of Digraphs, Embeddable in a Surface, without 4-Cycles

The positive lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the positive lightness of digraphs with nonzero minimum in-degree and a given girth k ⩾ 5 which can be embedded in a surface S. These bounds are tight for surfaces of nonnegative Euler characteristics. We further obtain bounds for classes of graphs where 3-cycles are permitted.