An improved bound for the crossing number of \(C_m\times C_n\): a self-contained proof using mostly combinatorial arguments
Let \(C_n\) denote the \(n\)-cycle graph, and let \(\times \) denote the Cartesian product of graphs. In 1973 Harary, Kainen, and Schwenk conjectured that the crossing number of \(C_n\times C_m\) for \(n\geq m\geq 3\) is \(n(m-2)\). This conjecture was recently proved if \(n\geq m(m+1)\), i.e. finitely many possible exceptions are left for every \(m\), see [\textit{L. Y. Glebsky} and \textit{G. Salazar}, J. Graph Theory 47, 53--72 (2004; Zbl 1053.05032)]. The current paper shows that for all \(m,n\) sufficiently large, almost 80 crossings are there.
scientific article; zbMATH DE number 2121763