The Join of Graphs and Crossing Numbers

It has been long–conjectured by Zarankiewicz that the crossing number of the complete bipartite graph K m , n equals ⌊ m − 1 2 ⌋ ⌊ m 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n 2 ⌋ . This conjecture has been verified by Kleitman for min { m , n } ⩽ 6 . Using these results, we give the exact values of crossing numbers for join of two paths, join of two cycles, and for join of path and cycle. In addition, we give the exact values of crossing numbers for join products G + P n and G + C n for all graphs G of order at most four.