On the 2-Colored Crossing Number

Abstract

Let D be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of D is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of D. First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph. To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices. We obtain such instances using a combination of heuristics and integer programming. Second, for any fixed drawing of K:n, we improve the bound on the ratio between its rectilinear 2-colored crossing number and its rectilinear crossing number.