Topology and Geometry of Crystallized Polyominoes

TL;DR

This paper completely characterizes the minimal tile count for polyominoes with a given number of holes, analyzing their structure, efficiency, and uniqueness using topological and dynamical methods.

Contribution

It introduces a complete solution to the minimal tile problem for polyominoes with holes, including structural properties, inequalities, and a new dynamical construction method.

Findings

Determined the minimal number of tiles g(h) for polyominoes with h holes.

Proved the uniqueness of certain crystallized polyominoes with specific numbers of holes.

Established a topological isoperimetric inequality relating perimeter, holes, and dual graph structure.

Abstract

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if it has $h$ holes and $g(h)$ tiles. We analyze structural properties of crystallized polyominoes and characterize their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. We also develop a new dynamical method of creating sequences of polyominoes which is invariant with respect to crystallization and efficient structure. Using this technique, we prove that crystallized polyominoes with $h_l=(2^{2l}-1)/3$ holes are unique.