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- W2749389382 abstract "Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Győri et al. showed that $frac{3}{8} le c le frac{2}{5}$. We prove that $c=frac{3}{8}$. More generally, we show that for any $varepsilon>0$, and any integer $k ge 2$, there is a $C_{2k}$-free graph $G_1$ which does not contain a bipartite subgraph of girth greater than $2k$ with more than $left(1-frac{1}{2^{2k-2}}right)frac{2}{2k-1}(1+varepsilon)$ fraction of the edges of $G_1$. There also exists a $C_{2k}$-free graph $G_2$ which does not contain a bipartite and $C_4$-free subgraph with more than $left(1-frac{1}{2^{k-1}}right)frac{1}{k-1}(1+varepsilon)$ fraction of the edges of $G_2$. One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős: For any $varepsilon>0$, and any integers $a$, $b$, $k ge 2$, there exists an $a$-uniform hypergraph $H$ of girth greater than $k$ which does not contain any $b$-colorable subhypergraph with more than $left(1-frac{1}{b^{a-1}}right)left(1+varepsilonright)$ fraction of the hyperedges of $H$. We also prove further generalizations of this theorem. In addition, we give a new and very short proof of a result of Kuhn and Osthus, which states that every bipartite $C_{2k}$-free graph $G$ contains a $C_{4}$-free subgraph with at least $1/(k-1)$ fraction of the edges of $G$. We also answer a question of Kuhn and Osthus about $C_{2k}$-free graphs obtained by pasting together $C_{2l}$'s (with $k>lge3$)." @default.
- W2749389382 created "2017-08-31" @default.
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- W2749389382 date "2017-08-17" @default.
- W2749389382 modified "2023-09-27" @default.
- W2749389382 title "On subgraphs of $C_{2k}$-free graphs and a problem of Kuhn and Osthus" @default.
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