Finding a biplanar imbedding of \(C_n\times C_n\times C_l\times P_m\). (Q2822814)

The biplanar crossing number of a graph \(G\) is the minimum sum of the crossing numbers of two graphs, into which the edge set of \(G\) can be partitioned. A graph is called biplanar if it has zero biplanar crossing number. \textit{É. Czabarka} et al. [Bolyai Soc. Math. Stud. 15, 55--77 (2006; Zbl 1098.05023)] proved that the Cartesian product of cycles \(C_k \times C_\ell \times C_m\) is biplanar, but \(C_k \times C_\ell \times C_m\times C_n\) is not. They asked for the biplanar crossing number of \(C_n \times C_n \times C_n \times P_n\), where \(P_n\) stands for an \(n\)-path. The paper under review answers this question in a more general form: \(C_n \times C_n \times C_\ell \times P_m\) is biplanar.