*Article* **Parity Properties of Configurations**

**Michal Staš**

Faculty of Electrical Engineering and Informatics, Technical University of Košice, 042 00 Košice, Slovakia; michal.stas@tuke.sk

**Abstract:** In the paper, the crossing number of the join product *G*<sup>∗</sup> + *Dn* for the disconnected graph *G*<sup>∗</sup> consisting of two components isomorphic to *K*<sup>2</sup> and *K*<sup>3</sup> 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*<sup>∗</sup> with paths *Pn* and cycles *Cn* on *n* vertices by adding new edges joining vertices of *Dn*.

**Keywords:** graph; join product; crossing number; configuration; parity properties; path; cycle

**MSC:** 05C10; 05C38
