The crossing number of \(C_6\times C_n\) (Q2712518)

The crossing number of \( C_m \times C_n \), \( 3 \leq m \leq n \), is known to be at most \( (m-2)n \). It is a long-standing conjecture that the crossing number of \( C_m \times C_n \) is, in fact, equal to this upper bound, and this conjecture has been proved for the pairs \( m=n=3,4,5,6,7 \), and \( m = 3,4,5 \) and arbitrary \( n \geq m \). In their main theorem, the authors prove the conjecture for the next unresolved case, namely, they show that \( cr(C_6 \times C_n) = 4n \), for all \( n \geq 6 \). The proof by induction is based on two theorems; the first dealing with the number of crossings of \( 6 \)-cycles in an ``optimal'' drawing of \( C_6 \times C_n \), and the second one (proved in another paper of the second author) dealing with the numbers of crossings of cycles in general \( C_m \times C_n \)'s. In their last section, the authors point out that the next case of \( m = 7 \) should be accessible using methods similar to those presented.