Intersections of curve systems and the crossing number of \(C_ 5\times C_ 5\) (Q1346134)

Let \({\mathcal C}_ 1\) and \({\mathcal C}_ 2\) be two families of pairwise disjoint simple closed curves in the plane such that each curve in \({\mathcal C}_ 1\) intersects each curve in \({\mathcal C}_ 2\). The authors provide that the total number of points of intersection in \({\mathcal C}_ 1 \cup {\mathcal C}_ 2\) is at least \(2(m - 1)n\), where \(m = | {\mathcal C}_ 1 |\) and \(n = | {\mathcal C}_ 2 |\). Moreover, this result has been applied to show that the Cartesian product of two 5-cycles has crossing number 15.