Edge disjoint Paths in Graphs on Surfaces

We present a survey of results on the edge disjoint paths problem and re late this problem to the edge disjoint homotopic path and the edge disjoint homotopic cycle problem The latter problem is given a graph G V E embedded on a surface S and closed curves C Ck on S nd necessary and su cient conditions for the existence of pairwise edge disjoint cycles e C e Ck in G so that e Ci is homotopic to Ci for i k We explain that a certain cut condition which is easily seen to be necessary is also suf cient for a fractional solution of the edge disjoint homotopic cycle problem To this end we use a theorem stating that any system of closed curves can be made minimally crossing by Reidemeister moves This establishes a relation between the edge disjoint homotopic cycle problem and the theory of knots