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arXiv · 2603.19038

Supercritical Site Percolation on Regular Graphs

Sahar Diskin·Michael Krivelevich·Itay Markbreit
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Abstract

We consider site (vertex) percolation on $d$-regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erd\H{o}s-R\'enyi component phenomenon" in the supercritical regime $p=\frac{1+\epsilon}{d-1}$: namely, the appearance of a unique giant component of order $n/d$ in the percolated subgraph, with all other components being of size $O(\log n)$. Our main results apply both to the $d$-dimensional hypercube and to pseudo-random graphs, and resolve two open questions in these cases. We further discuss differences (and similarities) between bond (edge) percolation setting and site percolation setting.

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Sahar Diskin, Michael Krivelevich, Itay Markbreit. 2026-03-19. Supercritical Site Percolation on Regular Graphs. https://arxiv.org/abs/2603.19038

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