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- W3196558748 abstract "Given a family $mathcal{F}$ of bipartite graphs, the {it Zarankiewicz number} $z(m,n,mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $mathcal{F}$ as a subgraph (such $G$ is called {it $mathcal{F}$-free}). For $1leq beta<alpha<2$, a family $mathcal{F}$ of bipartite graphs is $(alpha,beta)$-{it smooth} if for some $rho>0$ and every $mleq n$, $z(m,n,mathcal{F})=rho m n^{alpha-1}+O(n^beta)$. Motivated by their work on a conjecture of ErdH{o}s and Simonovits on compactness and a classic result of Andr'asfai, ErdH{o}s and S'os, in cite{AKSV} Allen, Keevash, Sudakov and Verstraete proved that for any $(alpha,beta)$-smooth family $mathcal{F}$, there exists $k_0$ such that for all odd $kgeq k_0$ and sufficiently large $n$, any $n$-vertex $mathcal{F}cup{C_k}$-free graph with minimum degree at least $rho(frac{2n}{5}+o(n))^{alpha-1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $delta>0$, there exists $k_0$ such that for all odd $kgeq k_0$ and sufficiently large $n$, any $n$-vertex $mathcal{F}cup{C_k}$-free graph with minimum degree at least $delta n^{alpha-1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $mathcal{F}$ consisting of the single graph $K_{s,t}$ when $tgg s$. We also prove an analogous result for $C_{2ell}$-free graphs for every $ellgeq 2$, which complements a result of Keevash, Sudakov and Verstraete in cite{KSV}." @default.
- W3196558748 created "2021-09-13" @default.
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- W3196558748 date "2021-09-03" @default.
- W3196558748 modified "2023-09-23" @default.
- W3196558748 title "Bipartite-ness under smooth conditions" @default.
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- W3196558748 doi "https://doi.org/10.48550/arxiv.2109.01311" @default.
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