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 $r$.
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...
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)...
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
For integers $n,r,t$ with $2\le t\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determi...
For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erd\H{o}s-S\'{o}s theorem, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1,$ $$\operatornam...
A graph is $K_{r+1}$-saturated if it is $K_{r+1}$-free and adding any missing edge creates a copy of $K_{r+1}$. Kim, Kim, Kostochka, and O conjectured that $K_{r-1}\vee(n-r+1)K_1$ minimizes the spectral radius among all $n$-vertex $K_{r+1}$-saturated graphs. They proved the case $r=2$, and the cases $r=3$ and $r\in\{4,...
Zheng-Bo Chen, Xiao-Dong Zhang· 0 citations
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