Colouring bottomless rectangles and arborescences

TL;DR

This paper investigates the limitations of semi-online colouring algorithms for bottomless rectangles and arborescences, revealing fundamental bounds and proposing improved bounds for polychromatic colourings.

Contribution

It establishes impossibility results for semi-online algorithms in colouring bottomless rectangles and arborescences, and provides new bounds for polychromatic colouring numbers in various rectangle families.

Findings

Semi-online algorithms cannot guarantee multi-colour coverage for bottomless rectangles.

Any semi-online arborescence colouring produces arbitrarily long monochromatic paths.

For certain rectangle families, the polychromatic k-colouring number is linear in k.

Abstract

We study problems related to colouring bottomless rectangles. One of our main results shows that for any positive integers $m, k$, there is no semi-online algorithm that can $k$-colour bottomless rectangles with disjoint boundaries in increasing order of their top sides, so that any $m$-fold covered point is covered by at least two colours. This is, surprisingly, a corollary of a stronger result for arborescence colourings. Any semi-online colouring algorithm that colours an arborescence in leaf-to-root order with a bounded number of colours produces arbitrarily long monochromatic paths. This is complemented by optimal upper bounds given by simple online colouring algorithms from other directions. Our other main results study configurations of bottomless rectangles in an attempt to improve the \textit{polychromatic $k$-colouring number}, $m_k^*$. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, $m_k^*$ is linear in $k$. We also present an improved lower bound for general families: $m_k^* \geq 2k-1$.