Maximizing the Degree of (Geometric) Thickness-t Regular Graphs

In this paper, we show that there exist $$6t-1$$-regular graphs with thickness t, by constructing such an example graph. Since all graphs of thickness t must have at least one node with degree less than 6t, this construction is optimal. We also show, by construction, that there exist 5t-regular graphs with geometric thickness at most t. Our construction for the latter builds off of a relationship between geometric thickness and the Cartesian product of two graphs.