The crossing numbers of join products of $K_4\cup K_1$ with cycles

Document Type : Original paper


Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Kosice


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.


Main Subjects

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