Polyominoes with maximal number of deep holes

TL;DR

This paper investigates the maximum number of deep holes in polyominoes, establishing bounds and asymptotic behavior, and computes exact values for specific cases, advancing understanding of polyomino extremal properties.

Contribution

The paper determines bounds and asymptotic behavior for the maximum number of deep holes in polyominoes, and computes exact values for an infinite subset of sizes.

Findings

Maximum number of deep holes is asymptotically n/3.

Bounds for the number of deep holes are established using Pick's theorem.

Exact values of deep holes are computed for specific polyomino sizes.

Abstract

In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers.