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 \(C_{n^2} \times P_{2m}\).