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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Algorithmic solution of extremal digraph problems


Authors: W. G. Brown, P. Erdős and M. Simonovits
Journal: Trans. Amer. Math. Soc. 292 (1985), 421-449
MSC: Primary 05C35; Secondary 05C20
DOI: https://doi.org/10.1090/S0002-9947-1985-0808730-0
MathSciNet review: 808730
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Abstract: For a given family $\mathcal {L}$ of digraphs, we study the "extremal" digraphs on $n$ vertices containing no member of $\mathcal {L}$, and having the maximum number of arcs, $\operatorname {ex} (n,\mathcal {L})$. We resolve conjectures concerning the set $\{ {\lim _{n \to \infty }}(\operatorname {ex} (n,\mathcal {L})/{n^2})\}$ as $\mathcal {L}$ ranges over all possible families, and describe a "finite" algorithm that can determine, for any $\mathcal {L}$, all matrices $A$ for which a sequence $\{ A(n)\}$ of "matrix digraphs" is asymptotically extremal ($A(n)$ contains no member of $\mathcal {L}$ and has $\operatorname {ex} (n,\mathcal {L}) + o({n^2})$ arcs as $n \to \infty$.) Résumé. Pour une famille donnée, $\mathcal {L}$, de digraphes, on étudie les digraphes "extrémaux" à $n$ sommets qui ne contiennent aucun membre de $\mathcal {L}$, et qui possèdent le nombre maximal d’arêtes, $\operatorname {ex} (n,\mathcal {L})$. On résolue des conjectures qui concernent l’ensemble $\{ {\lim _{n \to \infty }}(\operatorname {ex} (n,\mathcal {L})/{n^2})\}$ où $\mathcal {L}$ soit une famille quelconque, et on présente un algorithme "fini" qui peut déterminer, pour chaque $\mathcal {L}$, toute matrice $A$ pour laquelle une suite $\{ A(n)\}$ de "digraphes matriciels" est extrémale asymptotiquement ($A(n)$ ne contient aucun membre de $\mathcal {L}$ et possède $\operatorname {ex} (n,\mathcal {L}) + o({n^2})$ arêtes lorsque $n \to \infty$.)


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Article copyright: © Copyright 1985 American Mathematical Society