Primality of closed path polyominoes
Combinatorics
2022-08-30 v3 Commutative Algebra
Abstract
In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.
Cite
@article{arxiv.2006.13935,
title = {Primality of closed path polyominoes},
author = {Carmelo Cisto and Francesco Navarra},
journal= {arXiv preprint arXiv:2006.13935},
year = {2022}
}
Comments
To appear in Journal of Algebra and its Applications