Acta mathematica scientia,Series A

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ON THE CROSSING NUMBER OF THE COMPLETE TRIPARTITE GRAPH K1,8,n

Huang Yuanqiu; Zhao Tinglei   

  1. Department of Mathematics, Hunan Normal University, Changsha 410081, China
  • Received:2005-06-28 Revised:1900-01-01 Online:2006-12-25 Published:2006-12-25
  • Contact: Huang Yuanqiu

Abstract: The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤ n) is Z(m,n), where Z(m,n)=\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor\lfloor\frac{n}{2}\rfloor$$\lfloor\frac{n-1}{2}\rfloor$ (for any real number x, $\lfloor x\rfloor$ denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤ 6. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤ 9, then the crossing number of the complete tripartite graph K1,8,n is $Z(9, n)+ 12\lfloor\frac{n}{2}\rfloor$.

Key words: Graphs, Drawing, Crossing number, Complete tripartite graph, Complete tripartite graph

CLC Number: 

  • 05C10
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