More on Decomposing Coverings by Octants

TL;DR

This paper improves bounds on decomposing multiple coverings of points in space and plane by octants and triangles, and links these geometric problems to dynamic interval coloring.

Contribution

It establishes new upper and lower bounds for decomposing coverings by octants and triangles, and connects these geometric problems to dynamic interval coloring.

Findings

Every 9-fold covering by octants decomposes into two coverings.

There exists a 4-fold covering that cannot be decomposed into two coverings.

The bounds also apply to coverings by triangles and relate to interval coloring problems.

Abstract

In this note we improve our upper bound given earlier by showing that every 9-fold covering of a point set in the space by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4-fold covering that does not decompose into two coverings. The same bounds also hold for coverings of points in $\R^2$ by finitely many homothets or translates of a triangle. We also prove that certain dynamic interval coloring problems are equivalent to the above question.