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Primality of closed path polyominoes

Universitá di Messina, Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Viale Ferdinando Stagno D’Alcontres 31, 98166 Messina, Italy

E-mail Address: carmelo.cisto@unime.it

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Francesco Navarra

Universitá di Messina, Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Viale Ferdinando Stagno D’Alcontres 31, 98166 Messina, Italy

E-mail Address: francesco.navarra@unime.it

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https://doi.org/10.1142/S021949882350055XCited by:7 (Source: Crossref)

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.

Communicated by S. K. Jain

Keywords:

AMSC: 05B50, 05E40, 13C05, 13G05

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