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Summary: The Shannon Capacity of a union
Noga Alon
To the memory of Paul Erdos
Abstract
For an undirected graph G = (V, E), let Gn
denote the graph whose vertex set is V n
in which
two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent iff for all i between 1 and
n either ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn((Gn
))1/n
,
where (Gn
) is the maximum size of an independent set of vertices in Gn
. 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.
1 Introduction
For an undirected graph G = (V, E), let Gn denote the graph whose vertex set is V n in which two
distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent iff for all i between 1 and n either
ui = vi or uivi E. The Shannon capacity c(G) of G is the limit limn((Gn))1/n, where (Gn) is
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