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 Sawhney confirmed this conjecture for every constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$, where, in a recent paper, Ara\'ujo and Mattos confirmed the conjecture for $p \in (4n^{-1/2}, 1/2)$. This, together with a general result of R\"ollin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $n^{-1/m(H)}\ll p\leq 1/2$, essentially settling the conjecture for cliques.
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Let $T$ be a tree on $k$ vertices, and write $n=a(k-...
Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erd\H{o}s and Simonovits states that for any rational $\gamma\in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=\...
Tao Jiang, Sean Longbrake, Liana Yepremyan· 0 citations
In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\leq t\leq r-1$, any $r$-chromatic...
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 an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random Erd\H{o}s--R\'enyi bipartite graph $G_{\alpha}(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-S...
Deepesh Singhal· 2 citations
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