Colouring bottomless rectangles and arborescences

We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring number $m_k^{⁎}$. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, $m_k^{⁎}$ is linear in k. We present the lower bound $m_k^{⁎}\ge 2k-1$ for general families.

We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers $m,k$, there is no semi-online algorithm that can k-colour bottomless rectangles presented in that order, so that any m-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.