Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SUBTOURNAMENTS ISOMORPHIC TO W5 OF AN INDECOMPOSABLE TOURNAMENT

  • Belkhechine, Houmem (Departement De Mathematiques Institut Preparatoire Aux Etudes D'Ingenieurs De Bizerte Universite De Carthage) ;
  • Boudabbous, Imed (Departement De Mathematiques Institut Preparatoire Aux Etudes D'Ingenieurs De Sfax Universite De Sfax) ;
  • Hzami, Kaouthar (Departement De Mathematiques Faculte Des Sciences De Sfax Universite De Sfax)
  • Received : 2011.06.21
  • Published : 2012.11.01
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Abstract

We consider a tournament T = (V,A). For each subset X of V is associated the subtournament T(X) = (X,$A{\cap}(X{\times}X)$) of T induced by X. We say that a tournament T' embeds into a tournament T when T' is isomorphic to a subtournament of T. Otherwise, we say that T omits T'. A subset X of V is a clan of T provided that for a, $b{\in}X$ and $x{\in}V{\backslash}X$, $(a,x){\in}A$ if and only if $(b,x){\in}A$. For example, ${\emptyset}$, $\{x\}(x{\in}V)$ and V are clans of T, called trivial clans. A tournament is indecomposable if all its clans are trivial. In 2003, B. J. Latka characterized the class ${\tau}$ of indecomposable tournaments omitting a certain tournament $W_5$ on 5 vertices. In the case of an indecomposable tournament T, we will study the set $W_5$(T) of vertices $x{\in}V$ for which there exists a subset X of V such that $x{\in}X$ and T(X) is isomorphic to $W_5$. We prove the following: for any indecomposable tournament T, if $T{\notin}{\tau}$, then ${\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}$ -2 and ${\mid}W_5(T){\mid}{\geq}{\mid}V{\mid}$ -1 if ${\mid}V{\mid}$ is even. By giving examples, we also verify that this statement is optimal.

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References

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