%0 Journal Article
%T A New Proof on the Bipartite Turán Number of Bipartite Graphs
%A Shiqian Wang
%J Engineering
%P 301-308
%@ 1947-394X
%D 2024
%I Scientific Research Publishing
%R 10.4236/eng.2024.169022
%X The bipartite Tur&#225;n number of a graph <i>H</i>, denoted by <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>ex</mtext><mrow><mo>(</mo>
   <mrow>
    <mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>H</mi></mrow>
  <mo>)</mo></mrow></mrow>
</math>

, is the maximum number of edges in any bipartite graph <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>G</mi><mo>=</mo><mrow><mo>(</mo>
   <mrow>
    <mi>A</mi><mo>,</mo><mi>B</mi><mo>;</mo><mi>E</mi><mrow><mo>(</mo>
     <mi>G</mi>
    <mo>)</mo></mrow></mrow>
  <mo>)</mo></mrow></mrow>
</math>

 with <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>|</mo> <mi>A</mi> <mo>|</mo></mrow><mo>=</mo><mi>m</mi></mrow>
</math>

 and <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>|</mo> <mi>B</mi> <mo>|</mo></mrow><mo>=</mo><mi>n</mi></mrow>
</math>

 which does not contain <i>H</i> as a subgraph. When<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>min</mi><mrow><mo>{</mo> <mrow>
   <mi>m</mi><mo>,</mo><mi>n</mi></mrow> <mo>}</mo></mrow><mo>&#x003E;</mo><mn>2</mn><mi>t</mi></mrow>
</math>

, the problem of determining the value of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>ex</mtext><mrow><mo>(</mo>
   <mrow>
    <mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><msub>
     <mi>K</mi>
     <mrow>
      <mi>m</mi><mo>&#x2212;</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>&#x2212;</mo><mi>t</mi></mrow>
    </msub>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 has been solved by Balbuena <i>et al.</i> in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mtext>ex</mtext><mrow><mo>(</mo>
   <mrow>
    <mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><msub>
     <mi>K</mi>
     <mrow>
      <mi>m</mi><mo>&#x2212;</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>&#x2212;</mo><mi>t</mi></mrow>
    </msub>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mrow><mo>{</mo> <mrow>
   <mn>0</mn><mo>,</mo><mn>1</mn></mrow> <mo>}</mo></mrow></mrow>
</math>

-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].
%K Bipartite Tur&#225
%K n Number
%K Adjacency Matrix
%K Zarankiewicz Problem
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=136373