Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

Culbertson, Jared; Dochtermann, Anton; Guralnik, Dan P.; Stiller, Peter F.

doi:10.37236/9120

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…This theorem, along with results from [9], imply that for these complexes the notions of vertex decomposable, shellable, shelling completable, and extendably shellable are all equivalent.…”

Section: Introductionmentioning

confidence: 78%

“…In [9] it is shown that a d-dimensional complex ∆ on d + 3 vertices is extendably shellable if and only if ∆ is shellable. In this section we show that these conditions are also equivalent to ∆ being vertex decomposable.…”

Section: Complexes With Few Verticesmentioning

confidence: 99%

“…The k = 2 case of Simon's Conjecture follows from [6] by considering the uniform matroid of rank 3, and Bigdeli, Yazdan Pour, and Zaare-Nahandi [2] have established the k ≥ n − 3 cases (a simpler proof was provided independently by the second author [11] based on results of Culbertson, Guralnik, and Stiller [8]). In [9] it is shown that if ∆ is a d-dimensional simplicial complex on at most d + 3 vertices, then in fact the notions of shellable and extendably shellable are equivalent. This implies the k = n − 3 case of Simon's conjecture and also provides a generalization of Kleinschmidt's results.…”

Section: Introductionmentioning

confidence: 99%

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Completing and extending shellings of vertex decomposable complexes

Coleman¹,

Dochtermann²,

Nathan³

et al. 2020

Preprint

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We say that a pure d-dimensional simplicial complex ∆ on n vertices is shelling completable if ∆ can be realized as the initial sequence of some shelling of ∆ (d) n−1 , the d-skeleton of the (n − 1)dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable.In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if ∆ is a vertex decomposable complex then there exists an ordering of its ground set V such that adding the revlex smallest missing (d + 1)-subset of V results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as k-vertex decomposable complexes. We also show that if ∆ is a d-dimensional complex on at most d + 3 vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.

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“…This theorem, along with results from [9], imply that for these complexes the notions of vertex decomposable, shellable, shelling completable, and extendably shellable are all equivalent.…”

Section: Introductionmentioning

confidence: 78%

Section: Complexes With Few Verticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Completing and extending shellings of vertex decomposable complexes

Coleman¹,

Dochtermann²,

Nathan³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

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Embedding Dimensions of Simplicial Complexes on Few Vertices

et al. 2023

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We provide a simple characterization of simplicial complexes on few vertices that embed into the d-sphere. Namely, a simplicial complex on $$d+3$$ d + 3 vertices embeds into the d-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.

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