Covering rectangles by few monotonous polyominoes

TL;DR

This paper determines the minimum number of monotonous polyominoes needed to cover an m by n lattice rectangle, linking geometric covering problems with arrangements of lines on chessboards.

Contribution

It provides a precise formula for the minimal covering number of rectangles by monotonous polyominoes, a novel geometric result.

Findings

Exact formula for minimal covering number

Connection to line arrangements on chessboards

Advancement in geometric covering theory

Abstract

A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $f:[a,b] \to \mathbb{R}$ with $f(k) \notin \mathbb{Z}$ whenever $k \in \mathbb{Z}$. Our main result says that the least cardinality of a covering of a lattice $(m \times n)$-rectangle by monotonous polyominoes is $\left\lceil \frac{2}{3}\left(m+n-\sqrt{m^2+n^2-mn}\right)\right\rceil$. The paper is motivated by a problem on arrangements of straight lines on chessboards.