Primality of closed path polyominoes

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.

On Gröbner bases and Cohen-Macaulay property of closed path polyominoes

“…In [7] the authors support this conjecture introducing a new class of polyominoes, called closed paths, and showing that having no zig-zag walks is a necessary and sufficient condition for their primality. An analogous result is proved in [8] for the weakly closed path polyominoes. In [22] the authors study the reduced Gröbner basis of polyomino ideals and introduce some conditions in order to the generators of I P form the reduced Gröbner basis with respect to some suitable degree reverse lexicographic monomial orders.…”

“…, ℓ} with i = j, there exists no inner interval J of P such that z i , z j belong to V (J). In according to [7], we recall the definition of a closed path polyomino, and the configuration of cells characterizing its primality. We say that a polyomino P is a closed path if it is a sequence of cells A 1 , .…”

“…For instance, in Figure 3 there is a closed path having an L-configuration and a ladder of three steps. We recall that a closed path has no zig-zag walks if and only if it contains an L-configuration or a ladder of at least three steps (see [7]).…”

“…Many mathematicians have taken an interest in classifying all polyominoes, for which the quotient ring K[P] = S/I P is a normal Cohen-Macaulay domain. The primality of I P is studied in several papers, see [7], [8], [17], [18], [19], [21], [22], [25], [27], [28]. Moreover in [17] and [25] the authors prove that if P is a simple polyomino then K[P] is a normal Cohen-Macaulay domain.…”

“…It seems that the non-existence of zig-zag walks in a polyomino could characterize its primality [Conjecture 4.6,[21]]. In [7] the authors support this conjecture introducing a new class of polyominoes, called closed paths, and showing that having no zig-zag walks is a necessary and sufficient condition for their primality. An analogous result is proved in [8] for the weakly closed path polyominoes.…”