Finding a Bipartite Planar Embedding of $C_n\times C_n\times C_l\times P_m$

Abstract

Determining the biplanar crossing number of the graph $C_n\times C_n\times C_n\times P_n$ was a problem proposed in a paper by Czabarka, Sykora, Székely, and Vito [2]. We find as a corollary to the main theorem of this paper that the biplanar crossing number of the aforementioned graph is zero. This result follows from the decomposition of $C_n\times C_n\times C_n\times P_m$ into one copy of $C_{n^2}\times P_{lm},l-2$ copies of $C_{n^2}\times P_m$, and a copy of $C_{n^2}\times P_{2m}$.