Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 2 2024 06 01 The crossing numbers of join product of four graphs on six vertices with discrete graphs 241 252 14527 10.22049/cco.2023.27911.1393 EN Michal Staš Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University, 042 00 Košice, Slovak Republic Journal Article 2022 07 06 The main aim of the paper is to give the crossing number of the join product $G^\ast + D_n$ for the graph $G^\ast$ isomorphic to 4-regular graph on six vertices except for two distinct edges with no common vertex such that two remaining vertices are still adjacent, and where $D_n$ consists of $n$ isolated vertices. The proofs are done with possibility of an existence of a separating cycle in some particular drawing of the investigated graph $G^\ast$ and also with the help of well-known exact values for crossing numbers of join products of two subgraphs $H_k$ of $G^\ast$ with discrete graphs. https://comb-opt.azaruniv.ac.ir/article_14527_ab90f3f711824ade768067b0ac9ba2fb.pdf