Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 10 4 2025 12 01 The crossing numbers of join products of $K_4\cup K_1$ with cycles 933 948 14732 10.22049/cco.2024.28761.1706 EN Michal Staš Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University, 042 00 Košice, Slovak Republic Maria Timková Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University, 042 00 Košice, Slovak Republic Journal Article 2023 06 17 The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, we extend known results concerning crossing numbers of join products of two small graphs with cycles. The crossing number of the join product $G^\ast + C_n$ for the disconnected graph $G^\ast$ consisting of the complete graph $K_{4}$ and one isolated vertex is given, where $C_n$ is the cycle on $n$ vertices. The proof of the main result is done with the help of lemma whose proof is based on a special redrawing technique. Up to now, the crossing numbers of $G + C_n$ were done only for a few disconnected graphs $G$. Finally, by adding new edge to the graph $G^\ast$, we are able to obtain the crossing number of $G_1+C_n$ for one other graph $G_1$ of order five. https://comb-opt.azaruniv.ac.ir/article_14732_3ee57576d44ffa0d3a4f74d7b688f3fe.pdf