Keller's conjecture on cube tilings asserted that, in any tiling of $\mathbb{R}^d$ by unit cubes, there must exist two cubes that share a $(d-1)$-dimensional face. This is now known to be true in dimensions $d\leq 7$ and false for $d\geq 8$. In this article, we propose analogues of Keller's face-sharing property for integer tilings. We construct counterexamples to a ``strong" version of this property, and prove that a weaker version holds for integer tilings under appropriate additional assumptions.
@article{10_37236_13002,
author = {Benjamin Bruce and Izabella {\L}aba},
title = {Keller properties for integer tilings},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/13002},
zbl = {1561.05030},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13002/}
}
TY - JOUR
AU - Benjamin Bruce
AU - Izabella Łaba
TI - Keller properties for integer tilings
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/13002/
DO - 10.37236/13002
ID - 10_37236_13002
ER -
%0 Journal Article
%A Benjamin Bruce
%A Izabella Łaba
%T Keller properties for integer tilings
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/13002/
%R 10.37236/13002
%F 10_37236_13002
Benjamin Bruce; Izabella Łaba. Keller properties for integer tilings. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/13002
