Algorithmic Solution of Extremal Digraph Problems1

For a given family JC of digraphs, we study the "extremal" digraphs on n vertices containing no member of JC, and having the maximum number of arcs, e\(n,^f). We resolve conjectures concerning the set {lim,, ^x (ex(n,JC )/n2)) as JC ranges over all possible families, and describe a "finite" algorithm that can determine, for any JC, all matrices A for which a sequence {A(n)} of "matrix digraphs" is asymptotically extremal (A(n) contains no member of JC and has e\(n,JC) + o(«2) arcs as n -> oo.) Resume. Pour une famille donnee, JC, de digraphes, on etudie les digraphes "extremaux" a n sommets qui ne contiennent aucun membre de JC, et qui possedent le nombre maximal d'aretes, ex(n,JC). On resolue des conjectures qui concernent l'ensemble {lim„_oc (ex(«, JC)/n2)} oil JC soit une famille quelconque, et on presente un algorithme "fini" qui peut determiner, pour chaque JC, toute matrice A pour laquelle une suite {/((«)} de "digraphes matriciels" est extremale asymptotiquement (A(n) ne contient aucun membre de JC et possede ex(n,JC) + o(n2) aretes lorsque n -» oo.)