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arXiv · 2501.03404

On the upper tail of star counts in random graphs

Margarita Akhmejanova·Matas Šileikis
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Abstract

Let $X$ count the number of $r$-stars in the random binomial graph $\mathbb{G}(n,p)$. We determine, for fixed $r$ and $\varepsilon > 0$, the asymptotics of $\log \mathbb{P}(X \ge (1 + \varepsilon)\mathbb{E} X)$ assuming only $\mathbb{E} X \to \infty$ and $p \to 0$ thus giving a first class of irregular graphs for which the upper tail problem for subgraph counts (stated by Janson and Ruciński in 2004) is solved in the sparse setting.

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Margarita Akhmejanova, Matas Šileikis. 2025-01-29. On the upper tail of star counts in random graphs. https://arxiv.org/abs/2501.03404

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