Determining the biplanar crossing number of CkxClx CmxCn.

摘要

Determining the biplanar crossing number of the graph C n x Cn x C n x Pn was a problem proposed in a paper by Czabarka, Sykora, Szekely, and Vrt˘o [7]. We find as a corollary to an upcoming theorem of this dissertation that the biplanar crossing number of the aforementioned graph is zero. This result follows from decompositions of the graph Ck x Cl x Cm x Pn that lead to a biplanar imbedding. Thus, leading us to the generalization of the aforementioned open problem where we find the biplanar crossing number of Ck x Cl x Cm x Pn is zero.