Abstract
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 \(f\left( {m,{\theta _{k,s}}} \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {2k + 1} \right)/\left( {2k + 2} \right)}}} \right)\), which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write K ′k and K ′t,s for the subdivisions of Kk and Kt,s. We show that \(f\left( {m,K_k^\prime } \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {5k - 8} \right)/\left( {6k - 10} \right)}}} \right)\) and \(f\left( {m,K_{t,s}^\prime } \right) \geqslant {1 \over 2}m + \Omega \left( {{m^{\left( {5t - 1} \right)/\left( {6t - 2} \right)}}} \right)\), 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).
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Research partially supported by National Natural Science Foundation of China (Grant No. 12101124), The 13th Five-Year Plan of Fujian Province Science of Education (Grant No. FJJKCGZ19-245) and Foundation of Fujian University of Technology (Grant No. GY-Z20079).
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Lin, J. Maximum bipartite subgraphs in H-free graphs. Czech Math J 72, 621–635 (2022). https://doi.org/10.21136/CMJ.2022.0302-20
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DOI: https://doi.org/10.21136/CMJ.2022.0302-20