Zig-zag collection containment in every non-prime collection of cells

Prove that every collection of cells P whose inner 2-minor ideal I_P is not prime contains a zig-zag collection as a subcollection, namely a union of cell intervals supported by a sequence of inner intervals that intersect consecutively at single vertices and satisfy the edge-interval alignment conditions specified in Definition 3.1.

Background

The authors introduce zig-zag collections, a structured class of non-prime collections built from inner intervals that intersect at single vertices in a cyclic pattern. They use these to analyze Cohen–Macaulayness and Hilbert–Poincaré series of closed paths in the non-prime setting.

They explicitly refer to a conjecture (from prior work) positing that every non-prime collection of cells contains a zig-zag collection, suggesting a decomposition approach that could systematize the study of non-prime cases.

References

As conjectured in Conjecture 4.6, every non-prime collection of cells should contain a zig-zag collection, so the latter could be useful for gaining information on non-prime collections of cells, by applying a decomposition of them into suitable zig-zag collections.

— On Cohen-Macaulay non-prime collections of cells  (2401.09152 - Cisto et al., 2024) in Introduction

The forward direction of this conjecture was proven as Corollary 3.6 in , but its converse, which would give a sufficient condition for the primality of a polyomino, remains an open problem.

— Primality of dilated closed path polyominoes  (2609.30086 - Larney, 24 Sep 2026) in Section 2.4, subsection “Primality of non-simple polyominoes and the Zig-Zag Walk Conjecture”