On the upper tail of star counts in random graphs

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 Ruci\'nski in 2004) is solved in the sparse setting.