Title: | Maximum bipartite subgraphs in $H$-free graphs (English) |
Author: | Lin, Jing |
Language: | English |
Journal: | Czechoslovak Mathematical Journal |
ISSN: | 0011-4642 (print) |
ISSN: | 1572-9141 (online) |
Volume: | 72 |
Issue: | 3 |
Year: | 2022 |
Pages: | 621-635 |
Summary lang: | English |
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Category: | math |
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Summary: | Given a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. In this paper we prove that $f(m,\theta _{k,s})\geq \tfrac 12 m +\Omega (m^{(2k+1)/(2k+2)})$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K'_{k}$ and $K'_{t,s}$ for the subdivisions of $K_k$ and $K_{t,s}$. We show that $f(m,K'_{k})\geq \tfrac 12 m +\Omega (m^{(5k-8)/(6k-10)})$ and $f(m,K'_{t,s})\geq \tfrac 12 m +\Omega (m^{(5t-1)/(6t-2)})$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs. All of these contribute to a conjecture of N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov (2003). (English) |
Keyword: | bipartite subgraph |
Keyword: | $H$-free |
Keyword: | partition |
MSC: | 05C35 |
MSC: | 05C70 |
idZBL: | Zbl 07584091 |
idMR: | MR4467931 |
DOI: | 10.21136/CMJ.2022.0302-20 |
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Date available: | 2022-08-22T08:15:04Z |
Last updated: | 2022-12-27 |
Stable URL: | http://hdl.handle.net/10338.dmlcz/150605 |
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