A Survey on the Crossing Number Problem

The crossing number of a graph G, denoted by cr(G), is defined to be the least number of its edges' crossings that might arise among all its drawings in the plane. The origin of this concept goes back to the Hungarian mathematician Paul Turan in 1944, when he was forced to work in a labor camp during the World War II, his problem (known as Turan's brick factory problem) in graphic terms asks: what is the minimum number of crossings amongst the edges if the complete bipirtite graph is drawn in the plane?. In this book, we collect together the results and papers that dealt with the crossing number problem through four main sections. The first section includes the presentation of the conjectures that are still uncertain, arranged by date, from the oldest to newest, the second section contains the known values for the Cartesian product of two graphs, the third section concerns the known values for join product of two graphs and the last section shows the known values for other kinds of graphs. Furthermore, we exhibit interesting examples for each presented conjecture.