Lower Bounds for the Thickness and the Total Number of Edge Crossings of Euclidean Minimum Weight Laman Graphs and (2,2)-Tight Graphs

We explore the maximum total number of edge crossings and the maximum geometric thickness of the Euclidean minimum-weight (k, ℓ)-tight graph on a planar point set P. In this paper, we show that (10/7-ε)|P| and (11/6-ε)|P| are lower bounds for the maximum total number of edge crossings for any ε>0 in cases (k, ℓ)=(2, 3) and (2, 2), respectively. We also show that the lower bound for the maximum geometric thickness is 3 for both cases. In the proofs, we apply the method of arranging isomorphic units regularly. While the method is developed for the proof in case (k, ℓ)=(2, 3), it also works for different ℓ.