Improvement of the Crossing Lemma for the odd-crossing number

A graph is called $1$-planar if it can be drawn in the plane so that every edge is crossed by at most one other edge. It is known that the maximum number of edges of $1$-planar graphs is $4n-8$. We prove the following generalization. A graph is $1$-odd-planar if it can be drawn in the plane so that every edge is crossed {\em an odd number of times} by at most one other edge. We show that the maximum number of edges of $1$-odd-planar graphs is also $4n-8$. As a consequence, we improve the constant coefficient in the Crossing Lemma for the odd-crossing number.