Original Paper

Combinatorica

, Volume 18, Issue 3, pp 301-310

The Shannon Capacity of a Union

  • Noga AlonAffiliated withDepartment of Mathematics, Raymond; and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University; Tel Aviv, Israel; E-mail: noga@math.tau.ac.il

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For an undirected graph \(\), let \(\) denote the graph whose vertex set is \(\) in which two distinct vertices \(\) and \(\) are adjacent iff for all i between 1 and n either \(\) or \(\). The Shannon capacity c(G) of G is the limit \(\), where \(\) is the maximum size of an independent set of vertices in \(\). We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956.

AMS Subject Classification (1991) Classes:  05C35, 05D10, 94C15