The crossing number of the Cartesian product of paths with complete graphs

Abstract In this paper, we determine the crossing number of K m ∖ e by the construction method for m ≤ 12 and apply the zip product to obtain that c r ( K m □ P n ) = ( n − 1 ) c r ( K m + 2 ∖ e ) + 2 c r ( K m + 1 ) for n ≥ 1 . Furthermore, we have c r ( K m □ P n ) = 1 4 ⌊ m + 1 2 ⌋ ⌊ m − 1 2 ⌋ ⌊ m − 2 2 ⌋ ( n ⌊ m + 4 2 ⌋ + ⌊ m − 4 2 ⌋ ) for n ≥ 1 , 1 ≤ m ≤ 10 , which is consistent with Zheng’s conjecture for the crossing number of K m □ P n .