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 $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)) \frac{n^2}{16}$. Balogh and Liu (JCTB, 2014) disproved this conjecture and determined the asymptotic value of $f_4(n,\mathrm{ex}(n,K_3)+1)$. He, Ma, Ma and Ye (JCTB, 2023) later determined $f_{p+1}(n,\mathrm{ex}(n,K_p)+1)$ asymptotically for every $p\ge 3$, and asked for the value of $f_{p+1}(n,m)$ for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. In this paper, we answer their question asymptotically for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. We also determine the exact value of $f_3(n,m)$ for all $0\le m\le \mathrm{ex}(n,K_3)$ by a different method.
For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put \[ I_G(\sigma)=|E(G)\cap E(\sigma(G))|. \] Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In 1977 Erd\H{o}s asked whether $f(n,4)=2n-4$, observing that $K_{2,n-2}$ gives the upper bound. We prove that, for all sufficiently large $n$, \[ f(n,4)=2n-4. \] Equivalently, every sufficiently large $n$-vertex graph with at most $2n-5$ edges has a relabelling with at most three common edges. Our proof is inspired by the recent work of Fang and Hou on the Erd\H{o}s--Mullin five-edge intersection problem and builds on their core--buffer and absorption framework. The main additional ingredients are a growing high-degree core $C$ satisfying \[ |C|\Delta(G-C)=o(n), \] and a rigidity analysis of the equality case in the relevant first-moment estimate. This analysis shows that the only core--buffer configuration forcing four local common edges is of $K_{2,|C|}$ type; the strict bound $e(G)\leq2n-5$ then supplies a defect which breaks this configuration.
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erd\H{o}s--S\'{o}s conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more general conjecture: if $n=\alpha(t-1)+\beta$ and $0\le\beta\le t-2$, then the graph $\alpha K_{t-1}\sqcup K_\beta$ maximizes the number of $r$-cliques among all $n$-vertex $T_t$-free graphs for every $3\le r\le t-2$. We show that this conjecture holds for $T_t$ having at least $t-r$ leaves with a common parent, which contains the star case as a special case and recovers the sharp clique-counting result conjectured by Gan, Loh and Sudakov and proved by Chase and Chao and Dong. We also study the clique-spectral analogue. Under the same leaf-bunch condition, every $T_t$-free graph $G$ satisfies $\rho_r(G)\le\binom{t-2}{r-1}$, with equality, for $n\ge t-1$, if and only if $K_{t-1}$ is a component of $G$. Furthermore, we prove the conjecture for $r=t-d$ whenever $d\ge2$ and $t\ge d^2-d+3$, while the case $d=1$ is determined exactly for every $t$. For $d\ge2$ and $t\ge d^2-d+3$, every $T_t$-free graph $G$ satisfies $\rho_{t-d}(G)\le\rho_{t-d}(K_{t-1})$, with equality characterized by the presence of a $K_{t-1}$-component. Our method is designed for relatively large cliques. In the leaf-poor case, after deleting edges that lie in no $(t-d)$-clique, we study the intersection relation among $(t-d)$-cliques and show that its equivalence classes induce the nontrivial clique-supported components; furthermore, we show each non-trivial component has at most $t-1$\) vertices. In the complementary leaf-rich case, a leaf-bunch criterion reduces the clique-counting problem to the sharp bounded-maximum-degree clique theorem.
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 $n=a(k-1)+b$, where $0\leq b<k-1$. Recently, Gerbner and Palmer (Electron. J. Combin., 2026) proposed the following conjecture: for every $r\geq3$, the graph $aK_{k-1}\cup K_b$ maximizes the number of copies of $K_r$ among all $n$-vertex $T$-free graphs. In this paper, we verify their conjecture when $r=k-2$ or $r=k-3\geq5$. More precisely, we show that ${\rm ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r}$ and characterize all extremal graphs.
For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In his 1977 formulation of the problem, Erd\H{o}s discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erd\H{o}s's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.
Given a graph $G$ with degree sequence $d_{1},\ldots,d_{n}$ and a positive real number $p$, let $e_{p}(G)=\sum_{i=1}^{n} d_{i}^{p}$. For a fixed family of graphs $\mathcal F$, let $ex_{p}(n, \mathcal F)$ denote the maximum value of $e_{p}(G)$ over all $\mathcal F$-free graphs $G$ on $n$ vertices. In 2000, Caro and Yuster introduced the following Tur\'an-type problem: For a positive integer $p$ and a fixed graph $F$, determine $ex_{p}(n, F)$, and characterize the extremal graphs $G$ on $n$ vertices that attain $ex_p(n, F)$. Recently, Gao, Liu, Ma and Pikhurko proved that $ex_{p}(n, \mathcal F)=(\tau(\mathcal F)-1+o(1))n^p$ for real $p>\frac{1}{1-\alpha}$, where $\mathcal F$ is a degenerate family of graphs with classical Tur\'an number $ex(n, \mathcal F)=O(n^{1+\alpha})$ for some $\alpha\in[0,1)$, and $\tau(\mathcal F)$ is the minimum size of an independent vertex cover over all bipartite graphs $F\in\mathcal F$. Based on their method, we obtain a stability result for $ex_{p}(n, \mathcal F)$, and prove that all extremal graphs must contain the complete bipartite graph $K_{\tau(\mathcal F)-1,n-\tau(\mathcal F)+1}$ when $n$ is sufficiently large. Our results can be used to deduce all previously known results about $ex_{p}(n, F)$ when $F$ is a bipartite graph and $n$ is sufficiently large. We also obtain several new exact results for $ex_{p}(n, F)$, namely, when $F$ is an even cycle, a complete bipartite graph, a discrete hypercube, a caterpillar forest, and a spider forest.
We have previously determined the maximum-spread $K_{s, t}$-minor-free graph(s) on $n$ vertices when $n$ is sufficiently large, $2\le s\le t$, and $s=2$ or $t\ge \frac{3}{2}(s-3) + \frac{4}{s-1}$. In this sequel paper, we completely determine the maximum-spread $K_{s, t}$-minor-free graphs on $n$ vertices for $n$ sufficiently large and $2\le s\le t$. In all of the remaining cases, the extremal graph is unique and is of the form $(K_r \vee (s-1-r)K_1) \vee (\ell_r K_t \cup (n-s+1-t\ell_r)K_1)$, where $r$ is an integer determined by $s$ and $t$ and $\ell_r$ is an integer determined by $n, s, t,$ and $r$.
William Linz, Linyuan Lu, Zhiyu Wang· 0 citations
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