On the Cartesian products with crossing number three

Abstract

M. Klešc et al. characterized graphs $G_1$ and $G_2$ for which the crossing number of their Cartesian product $G_1\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{G}_2$ equals one or two. In this paper, their results are extended by giving the necessary and sufficient conditions for all pairs of graphs $G_1$ and $G_2$ for which the crossing number of their Cartesian product $G_1\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{G}_2$ equals three, if one of the graphs $G_1$ and $G_2$ is a cycle.