Reconstruction of slices of cubes

Marie-Charlotte Brandenburg; Juliana Curtis; Chiara Giardino; Chiara Meroni; Arnau Padrol

Home/Archive/arXiv

arXiv · 2610.09939

Reconstruction of slices of cubes

Marie-Charlotte Brandenburg·Juliana Curtis·Chiara Giardino·Chiara Meroni·Arnau Padrol

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that the combinatorial type of a slice of the cube by a hyperplane is completely determined by its graph, within the class of cube slices. We also show that such a result cannot extend to slices of arbitrary simple polytopes by constructing a simple polytope that has two combinatorially distinct slices with isomorphic graphs.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Marie-Charlotte Brandenburg, Juliana Curtis, Chiara Giardino, Chiara Meroni, Arnau Padrol. 2026-10-07. Reconstruction of slices of cubes. https://arxiv.org/abs/2610.09939

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hecke Triangle Groups and Dessins d'Enfants

We study finite-index subgroups of the Hecke triangle group $H_q$ of signature $(2,q,\infty)$ through the finite-edge tessellation, special polygons, and the associated dessins. The quotient finite-edge graph is a connected bipartite ribbon graph in which even vertices have degree $1$ or $2$ and odd vertices may have any degree $d\mid q$; geometrically, degree $d$ records a $d$-cluster and an elliptic stabilizer of order $q/d$. We determine two finite ambiguities that arise when one passes from this quotient graph to a special polygon. First, all tree diagrams obtained from a fixed ribbon graph are classified by the possible spanning-tree cuts modulo graph automorphisms. Second, at a proper cluster one must record which cyclic gap contains the omitted branches of the universal $q$-star; every such choice develops to a special polygon. Combining the two gives an explicit finite description of all polygon orbits over a fixed dessin, together with a separate count of normalized realizations in the fixed tessellation. Concrete examples for $H_5$, $H_6$, and $H_7$ illustrate the geometry, and two arithmetic corollaries concern level-$2$ cusp labels and Galois orbits of principal congruence dessins.

math.CO↗

arXiv2026

A new simple family of non-periodic tilings with square tiles

We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles. Both families are defined by simple local rules, and the proof of their non-periodicity is as simple as that of the non-periodicity of Robinson's tilings. We also relate the construction to the classical chair tiling: forgetting the decoration, our tilings map almost one-to-one onto the chair, our substitution is essentially a square root of the chair substitution, and our local rules are perfect, defining exactly the family of substitution tilings.

math.CO↗

arXiv2026

Partitioning perfect graphs into comparability graphs

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

math.CO↗