A Hanani-Tutte Theorem for Cycles

Given a drawing D of a graph G, we define the crossing number between any two cycles C₁,C₂ in D to be the number of crossings that involve at least one edge from each of C₁ and C₂ except the crossings between edges that are common to both cycles. We show that if the crossing number between every two cycles in G is even in a drawing of G on the plane, then there is a planar drawing of G. This result can be extended to arbitrary surfaces.