Corner Peeling in Cubical Subcomplexes of Integer Three-Space

Definitions and mathematical statement

Equip ${\mathbb{R}}^3$ with its standard unit cubulation. Its lattice graph has vertex set ${\mathbb{Z}}^3$, with an edge between two vertices precisely when their difference is one of the signed coordinate unit vectors. Let $G$ be a finite connected subgraph of this graph; $G$ need not be an induced subgraph. Define $X(G)$ to contain the vertices and edges of $G$, every standard unit square whose four vertices and four edges belong to $G$, and every standard unit three-dimensional cube whose eight vertices and twelve edges belong to $G$. Thus the one-skeleton of $X(G)$ is $G$.

For $S\subseteq V(G)$, define the induced cubical subcomplex by \[ X(G)[S]=\{\sigma\in X(G):\text{ every vertex of }\sigma\text{ belongs to }S\}. \] A cell of a cubical complex is maximal if it is not properly contained in another cell of that complex. A corner is a vertex contained in exactly one maximal cell; maximal cells here may have dimension zero, one, two, or three. A corner peeling is an ordering $v_1,\ldots,v_n$ of all vertices of $G$, where $n=|V(G)|$, such that, for every $1\leq i\leq n$, the vertex $v_i$ is a corner of \[ X(G)[\{v_1,\ldots,v_i\}]. \] Equivalently, the vertices can be deleted as corners in the order $v_n,\ldots,v_1$.

The supplied question assumes that $X(G)$ is simply connected and that every connected component of its intersection with a coordinate hyperplane is simply connected. There is an unresolved specification in this hypothesis: the supplied record does not define the intended family of coordinate hyperplanes. To display this dependence explicitly, put \[ H_{j,t}=\{x\in{\mathbb{R}}^3:x_j=t\},\qquad j\in\{1,2,3\},\quad t\in{\mathbb{R}}. \] For a specified family $\mathcal H$ of such planes, the question has the following precise form:

For every finite connected lattice subgraph $G$ as above, if $X(G)$ is simply connected and every connected component of $X(G)\cap H$ is simply connected for every $H\in\mathcal H$, must $G$ admit a corner peeling?

Intersections are taken in the geometric realization in ${\mathbb{R}}^3$. Empty intersections have no connected components and impose no condition. The record does not resolve whether $\mathcal H$ means just the three planes $H_{j,0}$, all integer translates, all real translates, or another specified family. The displayed question is therefore a formulation conditional on that missing choice, not a resolution of it. The supplied original excerpt ends before the conjectural assertion, so it cannot settle this qualification.

Short English statement

The question asks whether a finite connected subgraph of the three-dimensional integer lattice can be reduced by repeatedly deleting a vertex lying in exactly one maximal cube, square, edge, or isolated vertex, assuming its associated cubical complex and all prescribed plane-section components are simply connected. The intended family of planes needs clarification. The record labels the question open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006627$, title: Corner Peeling in Cubical Subcomplexes of Integer Three-Space; exact source problem code: catalogue entry.
• Supplied attribution: Problem 15, Corners in subcomplexes of ${\mathbb{Z}}^3$, V. Chepoi, from discussions with J. Chalopin and M. Kokkou. The supplied original excerpt defines the complex and corners but terminates at the incomplete text "A" before the conjectural statement.
• Supplied report citation: Median Geometry and Applications (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/8, pp. 487--540. https://doi.org/10.4171/owr/2026/8.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/53603.
• Supplied literature source, labelled "related 2026 preprint": https://arxiv.org/abs/2602.12894. No title or authors for this preprint are supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the assertion open and states that the cited 2026 work proves a theorem under an additional condition, rather than proving the conjecture. This is supplied dated curation, not an independently verified literature assessment; the additional condition is not specified in the record.
• The registry marks the statement corrected_verified and reports that the official report was checked, the question isolated, and notation repaired. These are supplied source-verification claims, not verification performed for this draft. The background describes the report as documenting workshop discussions and explicitly requests independent verification of time-sensitive status and cited claims.