The Shannon Capacity of a Union

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.

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Received: December 8, 1997

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Alon, N. The Shannon Capacity of a Union. Combinatorica 18, 301–310 (1998). https://doi.org/10.1007/PL00009824

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