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arXiv · 2409.18109

Canonical labelling of sparse random graphs

Oleg Verbitsky·Maksim Zhukovskii
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Abstract

We show that if $p=O(1/n)$, then the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with high probability admits a canonical labeling computable in time $O(n\log n)$. Combined with the previous results on the canonization of random graphs, this implies that $G(n,p)$ with high probability admits a polynomial-time canonical labeling whatever the edge probability function $p$. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of $G(n,p)$.

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Oleg Verbitsky, Maksim Zhukovskii. 2024-09-26. Canonical labelling of sparse random graphs. https://arxiv.org/abs/2409.18109

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