---
_id: '795'
abstract:
- lang: eng
  text: 'We introduce a common generalization of the strong Hanani–Tutte theorem and
    the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where
    every pair of independent edges crosses an even number of times, then G has a
    planar drawing preserving the rotation of each vertex whose incident edges cross
    each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte
    theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler
    proof.'
article_number: P3.18
article_processing_charge: No
article_type: original
author:
- first_name: Radoslav
  full_name: Fulek, Radoslav
  id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
  last_name: Fulek
  orcid: 0000-0001-8485-1774
- first_name: Jan
  full_name: Kynčl, Jan
  last_name: Kynčl
- first_name: Dömötör
  full_name: Pálvölgyi, Dömötör
  last_name: Pálvölgyi
citation:
  ama: Fulek R, Kynčl J, Pálvölgyi D. Unified Hanani Tutte theorem. <i>Electronic
    Journal of Combinatorics</i>. 2017;24(3). doi:<a href="https://doi.org/10.37236/6663">10.37236/6663</a>
  apa: Fulek, R., Kynčl, J., &#38; Pálvölgyi, D. (2017). Unified Hanani Tutte theorem.
    <i>Electronic Journal of Combinatorics</i>. International Press. <a href="https://doi.org/10.37236/6663">https://doi.org/10.37236/6663</a>
  chicago: Fulek, Radoslav, Jan Kynčl, and Dömötör Pálvölgyi. “Unified Hanani Tutte
    Theorem.” <i>Electronic Journal of Combinatorics</i>. International Press, 2017.
    <a href="https://doi.org/10.37236/6663">https://doi.org/10.37236/6663</a>.
  ieee: R. Fulek, J. Kynčl, and D. Pálvölgyi, “Unified Hanani Tutte theorem,” <i>Electronic
    Journal of Combinatorics</i>, vol. 24, no. 3. International Press, 2017.
  ista: Fulek R, Kynčl J, Pálvölgyi D. 2017. Unified Hanani Tutte theorem. Electronic
    Journal of Combinatorics. 24(3), P3.18.
  mla: Fulek, Radoslav, et al. “Unified Hanani Tutte Theorem.” <i>Electronic Journal
    of Combinatorics</i>, vol. 24, no. 3, P3.18, International Press, 2017, doi:<a
    href="https://doi.org/10.37236/6663">10.37236/6663</a>.
  short: R. Fulek, J. Kynčl, D. Pálvölgyi, Electronic Journal of Combinatorics 24
    (2017).
corr_author: '1'
date_created: 2018-12-11T11:48:32Z
date_published: 2017-07-28T00:00:00Z
date_updated: 2025-07-10T11:54:52Z
day: '28'
ddc:
- '000'
department:
- _id: UlWa
doi: 10.37236/6663
ec_funded: 1
file:
- access_level: open_access
  checksum: ef320cff0f062051e858f929be6a3581
  content_type: application/pdf
  creator: dernst
  date_created: 2019-01-18T14:04:08Z
  date_updated: 2020-07-14T12:48:06Z
  file_id: '5853'
  file_name: 2017_ElectrCombi_Fulek.pdf
  file_size: 236944
  relation: main_file
file_date_updated: 2020-07-14T12:48:06Z
has_accepted_license: '1'
intvolume: '        24'
issue: '3'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: Electronic Journal of Combinatorics
publication_identifier:
  issn:
  - 1077-8926
publication_status: published
publisher: International Press
publist_id: '6859'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Unified Hanani Tutte theorem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2017'
...
---
_id: '701'
abstract:
- lang: eng
  text: 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.
article_processing_charge: No
author:
- first_name: Jan
  full_name: Kynčl, Jan
  last_name: Kynčl
- first_name: Zuzana
  full_name: Patakova, Zuzana
  id: 48B57058-F248-11E8-B48F-1D18A9856A87
  last_name: Patakova
  orcid: 0000-0002-3975-1683
citation:
  ama: Kynčl J, Patakova Z. On the nonexistence of k reptile simplices in ℝ^3 and
    ℝ^4. <i>The Electronic Journal of Combinatorics</i>. 2017;24(3):1-44.
  apa: Kynčl, J., &#38; Patakova, Z. (2017). On the nonexistence of k reptile simplices
    in ℝ^3 and ℝ^4. <i>The Electronic Journal of Combinatorics</i>. International
    Press.
  chicago: Kynčl, Jan, and Zuzana Patakova. “On the Nonexistence of k Reptile Simplices
    in ℝ^3 and ℝ^4.” <i>The Electronic Journal of Combinatorics</i>. International
    Press, 2017.
  ieee: J. Kynčl and Z. Patakova, “On the nonexistence of k reptile simplices in ℝ^3
    and ℝ^4,” <i>The Electronic Journal of Combinatorics</i>, vol. 24, no. 3. International
    Press, pp. 1–44, 2017.
  ista: Kynčl J, Patakova Z. 2017. On the nonexistence of k reptile simplices in ℝ^3
    and ℝ^4. The Electronic Journal of Combinatorics. 24(3), 1–44.
  mla: Kynčl, Jan, and Zuzana Patakova. “On the Nonexistence of k Reptile Simplices
    in ℝ^3 and ℝ^4.” <i>The Electronic Journal of Combinatorics</i>, vol. 24, no.
    3, International Press, 2017, pp. 1–44.
  short: J. Kynčl, Z. Patakova, The Electronic Journal of Combinatorics 24 (2017)
    1–44.
corr_author: '1'
date_created: 2018-12-11T11:48:00Z
date_published: 2017-07-14T00:00:00Z
date_updated: 2025-07-10T11:54:09Z
day: '14'
ddc:
- '500'
department:
- _id: UlWa
file:
- access_level: open_access
  checksum: a431e573e31df13bc0f66de3061006ec
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:14:25Z
  date_updated: 2020-07-14T12:47:47Z
  file_id: '5077'
  file_name: IST-2018-984-v1+1_Patakova_on_the_nonexistence_of_k-reptile_simplices_in_R_3_and_R_4_2017.pdf
  file_size: 544042
  relation: main_file
file_date_updated: 2020-07-14T12:47:47Z
has_accepted_license: '1'
intvolume: '        24'
issue: '3'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Submitted Version
page: 1-44
publication: The Electronic Journal of Combinatorics
publication_identifier:
  issn:
  - 1077-8926
publication_status: published
publisher: International Press
publist_id: '6996'
pubrep_id: '984'
quality_controlled: '1'
status: public
title: On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2017'
...