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Non-ridge-chordal complexes whose clique complex has shellable Alexander dual. (English) Zbl 1459.05355

Summary: A recent conjecture that appeared in three papers by M. Bigdeli and S. Faridi [ibid. 172, Article ID 105204, 33 p. (2020; Zbl 1439.13053)], A. Dochtermann [ibid. 177, Article ID 105327, 22 p. (2021; Zbl 1448.05236)], and A. Nikseresht [ibid. 168, 318–337 (2019; Zbl 1421.05101)], is that every simplicial complex whose clique complex has shellable Alexander dual, is ridge-chordal. This strengthens the long-standing Simon’s conjecture that the \(k\)-skeleton of the simplex is extendably shellable, for any \(k\). We show that the stronger conjecture has a negative answer, by exhibiting an infinite family of counterexamples.

MSC:

05E45 Combinatorial aspects of simplicial complexes
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05E40 Combinatorial aspects of commutative algebra
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References:

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