Squarability of rectangle arrangements
  We study when an arrangement of axis-aligned rectangles can be transformed into an arrangement of axis-aligned squares in $\mathbb{R}^2$ while preserving its structure. We found a counterexample to the conjecture of J. Klawitter, M. N\"ollenburg and T. Ueckerdt whether all arrangements without crossing and side-piercing can be squared. Our counterexample also works in a more general case when we only need to preserve the intersection graph and we forbid side-piercing between squares. We also show counterexamples for transforming box arrangements into combinatorially equivalent hypercube arrangements. Finally, we introduce a linear program deciding whether an arrangement of rectangles can be squared in a more restrictive version where the order of all sides is preserved. 
Mat\v{e}j Kone\v{c}n\'y, Stanislav Ku\v{c}era, Michal Opler, Jakub   Sosnovec, \v{S}t\v{e}p\'an \v{S}imsa, Martin T\"opfer
cs.CG