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arXiv · 2407.02058

Isoperimetry in product graphs

Sahar Diskin·Wojciech Samotij
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Abstract

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 Bollob\'as$\unicode{x2013}$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.

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Sahar Diskin, Wojciech Samotij. 2024-07-02. Isoperimetry in product graphs. https://arxiv.org/abs/2407.02058

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