We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $\Delta$, the binomial random graph $G(n,C\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $\Delta$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\ln n/n\leq p=n^{-1+o(1)}$, the minimum girth $\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contains all cycle factors of girth at least $\Omega(\ell)$. In particular, with high probability $G(n,C\ln n/n)$ contains all cycle factors of girth at least $100\ln n/\ln\ln n$, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transaction of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph an...
For a graph $G$ of order $n$, put $\sigma(G)=(\lambda_1(G)+\lambda_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maximize...
For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\a...
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1}...
Let $\zeta(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erd\H{o}s and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability. We resolve this problem along the full...
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
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