Diierent Capacities of a Digraph Jj Anos Kk Orner

The Shannon capacity of every induced subgraph of a perfect graph equals its clique number. However, for the co{normal powers of an odd cycle the super{multiplicativity of the clique number makes the determination of capacity one of the hardest problems in combinatorics. We study the asymptotic growth of induced subgraphs of some particular type in powers of a xed graph. In case of a simple graph we investigate the asymptotic growth in its co{normal powers of the largest induced subgraph on which non{adjacency is an equivalence relation, (equivalence graphs). For directed graphs we introduce a natural generalization of the co{normal product called Sperner product and analyze the directed version of the problem of asymptotic growth both for the clique number (Sperner capacity) and for induced subgraphs corresponding to equivalence{subgraphs with a particular direction of their edges, called waterfalls. A waterfall is an equivalence graph with linearly ordered equivalence classes in which every edge is present with a direction pointing away from the class which comes rst in the order. We prove for several classes of graphs that the maximum cardinality of a waterfall is multiplicative on Sperner powers although this is not true in general. For every class of digraphs closed under Sperner product, the cardinality of the largest subgraph belonging to the given class that is contained as an induced subgraph in the co{ normal powers of a graph G has an exponential growth. The corresponding asymptotic exponent is the capacity of G with respect to said class of digraphs. We derive upper and lower bounds for these capacities, and analyze the conditions under which they are tight.