On the crossing number of cm × cn

We show that the M-crossing number cr_M(C_m × C_n) of C_m × C_n behaves asymptotically according to lim_n→∞ {cr_M(C_m × C_n)/((m − 2)n)} = 1, for each m ≥ 3. This result reinforces the conjecture cr(C_m × C_n) = (m − 2)n if 3 ≤ m ≤ n, which has been proved only for m ≤ 6. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 163–170, 1998