Minimal-Perimeter Lattice Animals and the Constant-Isomer Conjecture

TL;DR

This paper investigates minimal-perimeter lattice animals on square and hexagonal lattices, establishing conditions under which inflating these animals produces all minimal-perimeter animals of larger size, and characterizes exceptional sizes.

Contribution

It provides a set of conditions ensuring inflation of minimal-perimeter animals generates all larger minimal-perimeter animals on certain lattices, advancing understanding of their structure.

Findings

Inflation of minimal-perimeter animals produces all larger minimal-perimeter animals under certain conditions.

Characterization of minimal-perimeter animals not generated by inflation.

Results demonstrated on square and hexagonal lattices.

Abstract

We consider minimal-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimal-perimeter animals of a certain size yields (without repetitions) all minimal-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimal-perimeter animals on these lattices that are not created by inflating members of another set of minimal-perimeter animals.