Covering Partial Cubes with Zones

Cardinal, Jean; Felsner, Stefan

doi:10.1007/978-3-319-13287-7_1

Cited by 4 publications

(7 citation statements)

References 14 publications

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“…Proof. This proof generalizes the proof idea for graphs presented in [CF18]. Since the convex hull and the Minkowski sum commute,…”

supporting

confidence: 52%

“…The following description of the vertices of the hypergraphic polytope in terms of acyclic orientations plays a central role in the remainder of this paper. This result (Proposition 4.8) is stated without proof in [CF18], where Cardinal and Felsner give a graph-theoretic proof for the vertex description of graphic polytopes. We adapt and generalize the proof idea to the hypergraphic setting.…”

Section: Applicationsmentioning

confidence: 98%

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Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra

Rehberg¹

2021

Preprint

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Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky ( 2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017) and Billera-Jia-Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.

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“…Proof. This proof generalizes the proof idea for graphs presented in [CF18]. Since the convex hull and the Minkowski sum commute,…”

supporting

confidence: 52%

Section: Applicationsmentioning

confidence: 98%

Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra

Rehberg¹

2021

Preprint

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show abstract

“…Proof. Consider the m-dimensional grid [3] m in R m and project it onto R 2 using a generic projection; that is, so that three points in the projection are collinear if and only if their preimages in [3] m are collinear. Denote by P the resulting planar point set and let n = 3 m .…”

Section: Subsets In General Position and The Hales-jewett Theoremmentioning

confidence: 99%

“…Denote by P the resulting planar point set and let n = 3 m . Since the projection is generic, the only collinear subsets of P are projections of collinear points in the original m-dimensional grid, and [3] m contains at most three collinear points. From Theorem 2.1, the largest subset of P with no three collinear points has size at most the indicated upper bound.…”

Section: Subsets In General Position and The Hales-jewett Theoremmentioning

confidence: 99%

General position subsets and independent hyperplanes in d-space

2016

Self Cite

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Abstract.Erdős asked what is the maximum number α(n) such that every set of n points in the plane with no four on a line contains α(n) points in general position. We consider variants of this question for d-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed d:• Every set H of n hyperplanes in R d contains a subset S ⊆ H of size at least c (n log n) 1/d , for some constant c = c(d) > 0, such that no cell of the arrangement of H is bounded by hyperplanes of S only.• Every set of cq d log q points in R d , for some constant c = c(d) > 0, contains a subset of q cohyperplanar points or q points in general position. Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013].

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“…There has been much theory developed about partial cubes, we direct an interested reader to books [11,17] and the survey [26]. For recent results in the field, see [1,8,9,15,28].…”

Section: Introductionmentioning

confidence: 99%

There are no finite partial cubes of girth more than 6 and minimum degree at least 3

Marc

2016

European Journal of Combinatorics

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Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube G with girth more than 6 is a tree-zone graph and therefore 2n(G) − m(G) − i(G) + ce(G) = 2 holds, where i(G) is the isometric dimension of G and ce(G) its convex excess.

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Covering Partial Cubes with Zones

Cited by 4 publications

References 14 publications

Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra

Pruned inside-out polytopes, combinatorial reciprocity theorems and generalized permutahedra

General position subsets and independent hyperplanes in d-space

There are no finite partial cubes of girth more than 6 and minimum degree at least 3

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