Mathematics – Combinatorics
Scientific paper
2008-01-26
Mathematics
Combinatorics
25 pages, presented at CGCS 2007(Luminy, France, May 2-4 2007) in honor of Michel Deza
Scientific paper
A tournament is \emph{acyclically indecomposable} if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The {\it profile} of a tournament $T$ is the function $\phi_T$ which counts for each integer $n$ the number $\phi_T(n)$ of tournaments induced by $T$ on the $n$-element subsets of $T$, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of $\phi_T$ is either polynomial, in which case $\phi_T(n)\simeq an^k$, for some positive real $a$, some non-negative integer $k$, or as fast as some exponential.
Boudabbous Youssef
Pouzet Maurice
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