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Polyominoes with maximal number of deep holes

Djordje Baralic, Shiven Uppal

TL;DR

The paper addresses the problem of determining the maximal number of deep holes that can be enclosed by an $n$-omino, denoted $h_n$. It combines constructive lower-bound techniques with a generalized Pick's theorem-based upper bound to establish $h_n = \frac{n}{3} + o(n)$ asymptotically, and computes $h_n$ exactly for an infinite subset of $n$ via detailed constructions and case analysis. The key contributions are tight asymptotic results, explicit lower bounds, and exact values for families of $n$, along with a framework that yields precise upper bounds. This advances understanding of deep-hole capacity in lattice-based polyominoes and informs related tiling/topology problems on planar grids.

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.

Polyominoes with maximal number of deep holes

TL;DR

The paper addresses the problem of determining the maximal number of deep holes that can be enclosed by an -omino, denoted . It combines constructive lower-bound techniques with a generalized Pick's theorem-based upper bound to establish asymptotically, and computes exactly for an infinite subset of via detailed constructions and case analysis. The key contributions are tight asymptotic results, explicit lower bounds, and exact values for families of , along with a framework that yields precise upper bounds. This advances understanding of deep-hole capacity in lattice-based polyominoes and informs related tiling/topology problems on planar grids.

Abstract

In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, of deep holes that an -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 -tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for , and show that asymptotically. To further develop these results, we compute as a function of for an infinite subset of positive integers.
Paper Structure (6 sections, 29 theorems, 58 equations, 6 figures)

This paper contains 6 sections, 29 theorems, 58 equations, 6 figures.

Table of Contents

  1. introduction
  2. Holes and deep holes
  3. Lower Bound
  4. Upper Bound
  5. Asymptotics of $h_n$
  6. Exact Value

Key Result

Proposition 3.1

For $n=(12 a^2+20a +8)+k$, where $k$ is an integer such that $0 \leq k < 24a + 32$, it holds In particular, if $k\geq 9$ then

Figures (6)

  • Figure 1: Tetris shapes
  • Figure 2: Free hexomino and heptomino, and a heptomino with one hole
  • Figure 3: Polyominoes with no deep holes and two deep holes
  • Figure 4: $ab$ deep holes enclosed by a $(3ab + 2a + 2b + 1)$-omino
  • Figure 5: Polyminoes enclosing $\left[\frac{n-3}{5}\right]$ holes for $n\equiv 2 \pmod{5}$
  • ...and 1 more figures

Theorems & Definitions (34)

  • Proposition 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • Proposition 6.1
  • ...and 24 more