Parity Properties of Configurations

Abstract

In the paper, the crossing number of the join product G*+Dn for the disconnected graph G* consisting of two components isomorphic to K2 and K3 is given, where Dn consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G*. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G* with paths Pn and cycles Cn on n vertices by adding new edges joining vertices of Dn.