The crossing number of <i>C_m</i> × <i>C_n</i> is as conjectured for <i>n</i> ≥ <i>m</i>(<i>m</i> + 1)

Abstract

Abstract It has been long conjectured that the crossing number of C m × C n is ( m −2) n , for all m , n such that n ≥ m ≥ 3. In this paper, it is shown that if n ≥ m ( m + 1) and m ≥ 3, then this conjecture holds. That is, the crossing number of C m × C n is as conjectured for all but finitely many n , for each m . The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and further developed by Adamsson and Richter. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 53–72, 2004