@inproceedings{184,
abstract = {We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.},
author = {Goaoc, Xavier and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
location = {Budapest, Hungary},
pages = {41:1 -- 41:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Shellability is NP-complete}},
doi = {10.4230/LIPIcs.SoCG.2018.41},
volume = {99},
year = {2018},
}
@article{701,
abstract = {A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2.},
author = {Kynčl, Jan and Patakova, Zuzana},
issn = {10778926},
journal = {The Electronic Journal of Combinatorics},
number = {3},
pages = {1--44},
publisher = {International Press},
title = {{On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4}},
volume = {24},
year = {2017},
}
@article{610,
abstract = {The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1 2k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.},
author = {Goaoc, Xavier and Mabillard, Isaac and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
journal = {Israel Journal of Mathematics},
number = {2},
pages = {841 -- 866},
publisher = {Springer},
title = {{On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result}},
doi = {10.1007/s11856-017-1607-7},
volume = {222},
year = {2017},
}
@inproceedings{1511,
abstract = {The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.},
author = {Goaoc, Xavier and Mabillard, Isaac and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
location = {Eindhoven, Netherlands},
pages = {476 -- 490},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result}},
doi = {10.4230/LIPIcs.SOCG.2015.476},
volume = {34 },
year = {2015},
}