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...
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 genera...
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-...
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 formulati...
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 Yust...
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 $n$ suffi...
William Linz, Lin-Yuan Lu, Zhi-Yu Wang· 0 citations
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