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We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A {em topological graph} T=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersetion point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is {em simple}, if any two edges meet in at most one common point. Let h=h(n) be the smallest integer such that every simple complete topological graph on n vertices contains an edge crossing at most h other edges. We show that Omega(n^{3/2})le h(n) le O(n^2/log^{1/4}n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn^2.\u

On Edges Crossing Few Other Edges in Simple Topological Complete Graphs