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

In this paper, we determine the crossing number of Km∖eKm∖e by the construction method for m≤12m≤12 and apply the zip product to obtain that cr(Km□Pn)=(n−1)cr(Km+2∖e)+2cr(Km+1)cr(Km□Pn)=(n−1)cr(Km+2∖e)+2cr(Km+1) for n≥1n≥1. Furthermore, we have cr(Km□Pn)=14⌊m+12⌋⌊m−12⌋⌊m−22⌋(n⌊m+42⌋+⌊m−42⌋) for n≥1n≥1, 1≤m≤101≤m≤10, which is consistent with Zheng’s conjecture for the crossing number of Km□PnKm□Pn.