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<p>The bipartite Turán number of a bipartite graph <i>H</i>, denoted by ex(<i>m, n; H</i>), is the maximum number of edges in subgraphs of <i>K</i><sub><i>m,n</i></sub> containing no copies of <i>H</i>. In this paper, we determine the bipartite Turán number of the complete bipartite graphs ex(<i>m; n; K</i><sub><i>s,t</i></sub>) = <i>mn</i> − (⌊(<i>n</i> − <i>t</i>)/<i>s</i>⌋(<i>m</i> − <i>s</i>) + <i>m</i> + <i>n</i> − <i>s</i> − <i>t</i> + 1) when <i>n</i> ≥ <i>s</i> + <i>t</i> and <i>t</i> ≥ <i>s</i>, if one of the following is satisfied: <i>n</i> = <i>s</i> + <i>t</i>; <i>m</i> ≤ <i>s</i> + (<i>t</i> − 1)/(⌊(<i>n</i> − <i>t</i>)/<i>s</i>⌋ + 1); <i>s</i>∣(<i>n</i> − <i>t</i>) and <i>m</i> ≤ <i>sn</i>/(<i>n</i> − <i>t</i>); being <i>m</i> ≥ 2<i>s</i> if <i>m</i> = <i>sn</i>/(<i>n</i> − <i>t</i>).</p>

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Bipartite Turán Problem on Complete Bipartite Graphs

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Lan-xin Wang,
  • Xia Zhang

摘要

The bipartite Turán number of a bipartite graph H, denoted by ex(m, n; H), is the maximum number of edges in subgraphs of Km,n containing no copies of H. In this paper, we determine the bipartite Turán number of the complete bipartite graphs ex(m; n; Ks,t) = mn − (⌊(n − t)/s⌋(m − s) + m + n − s − t + 1) when n ≥ s + t and t ≥ s, if one of the following is satisfied: n = s + t; m ≤ s + (t − 1)/(⌊(n − t)/s⌋ + 1); s∣(n − t) and m ≤ sn/(n − t); being m ≥ 2s if m = sn/(n − t).