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arXiv · 2606.18321

Proofs of Two Conjectures of Alon on Subgraph Counts

Peiru Kuang·Shuang Sun·Yan Wang·Jiasheng Zeng
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Abstract

All graphs considered are finite with no isolated vertices. Let $N(m,H)$ be the maximum number of subgraphs of a graph $G$ isomorphic to $H$, taken over all graphs $G$ with $m$ edges. Alon proved that $N(m,H)=\Theta_H(m^{\gamma(H)})$, where $\gamma(H)=(|V(H)|+D(H))/2$ and $D(H)=\max_{S\subseteq V(H)}(|S|-|N_H(S)|)$, and conjectured [Conjecture 1, Isr. J. Math., 1986] that limit of $N(m,H)/m^{\gamma(H)}$ exists as $m\to\infty$. We prove this conjecture and identify the limit as $\lambda(H)=\Lambda(H)/|\operatorname{Aut}(H)|$, where $\Lambda(H)$ is characterized by a variational problem over finite cores. We also resolve another conjecture of Alon [Conjecture 2, Isr. J. Math., 1986], which stated that if $H$ is a disjoint union of stars, then for every $m$ an extremal graph attaining $N(m,H)$ may be chosen to be a disjoint union of stars.

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Peiru Kuang, Shuang Sun, Yan Wang, Jiasheng Zeng. 2026-06-16. Proofs of Two Conjectures of Alon on Subgraph Counts. https://arxiv.org/abs/2606.18321

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