We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie-Weiss-type. We provide a criterion on the correlations ensuring the validity of Wigner's semicircle law for the eigenvalue distribution measure. In case of CurieWeiss distributions this criterion applies above the critical temperature (i. e. β < 1). We also investigate the largest eigenvalue of certain ensembles of Curie-Weiss type and find a transition in its behavior at the critical temperature.

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We consider the problem of allocating the cost of an optirnal traveling salesman tour in a fair way among the nodes visited; in particular, we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality. We thereby use the model of TSP games in the sense of cooperative game theory. We give examples showing that the core of such games may be empty, even for the case of Euclidean distances. On the positive, we develop an LP-based allocation rule guaranteeing that no coalition pays more than c~ times its own cost, where a is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation, which is also known as the solution over the "subtour polytope". A well-known conjecture states that c~ < 4-5-We also exhibit examples showing that this ratio cannot be improved below 4Summary. Wir betrachten die Aufgabe, die Kosten einer optimalen Traveling-Salesman-Tour fair unter den besuchten Knoten zu verteilen; insbesondere untersuchen wir den Fall, dab die Kostenmatrix des zugrundeliegenden TSP-Problems die Dreiecksungleichung erRillt. Dazu wird das Modell von TSP-Spielen im Sinne der kooperativen Spieltheorie benutzt. Wir zeigen anhand eines Beispiels, dab der Core eines solchen Spiels leer sein kann, selbst im Falle euklidischer Distanzen. Andererseits geben wit eine LP-basierte Verteilungsregel an, die garantiert, dag keine Koalition mehr als das c~-fache ihrer eigenen Kosten bezaahlen mug, wobei c~ das Verh/iltnis zwischen den Kosten einer optimalen TSP-Tour und dem Optimum der Held-Karp-Relaxation ist, die auch als L6sung fiber dem "subtour polytope" bekannt ist.4 Abschliegend Es wird allgemein vermutet, dab ct _< 5" geben wir eine Klasse yon Beispielen an, die beweist, dab keine allgemeine Verteilungsregel ftir das TSP-game ein

The nucleon is introduced as a new allocation concept for non-negative cooperative n-person transferable utility games. The nucleon may be viewed as the multiplicative analogue of Schmeidler's nucleolus. It is shown that the nucleon of (not necessarily bipartite) matching games can be computed in polynomial time.

The clique number ω(D) of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number χ(H) defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number ω(H). Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐scriptNP‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is NP‐complete to decide whether a perfect digraph has a kernel.