In this short note, we establish an edge-isoperimetric inequality for arbitrary product graphs. Our inequality is sharp for subsets of many different sizes in every product graph. In particular, it implies that the $2^d$-element sets with smallest edge-boundary in the hypercube are subcubes and is only marginally weaker than the Bollobás-Leader edge-isoperimetric inequalities for grids and tori. Additionally, it improves two edge-isoperimetric inequalities for products of regular graphs proved by Erde, Kang, Krivelevich, and the first author and answers two questions about edge-isoperimetry in powers of regular graphs raised in their work.
@article{10_37236_13585,
author = {Sahar Diskin and Wojciech Samotij},
title = {Isoperimetry in product graphs},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/13585},
zbl = {8097640},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13585/}
}
TY - JOUR
AU - Sahar Diskin
AU - Wojciech Samotij
TI - Isoperimetry in product graphs
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/13585/
DO - 10.37236/13585
ID - 10_37236_13585
ER -
%0 Journal Article
%A Sahar Diskin
%A Wojciech Samotij
%T Isoperimetry in product graphs
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/13585/
%R 10.37236/13585
%F 10_37236_13585
Sahar Diskin; Wojciech Samotij. Isoperimetry in product graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/13585
