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The Shannon capacity of a graph and the independence numbers of Eyal Lubetzky

Summary: The Shannon capacity of a graph and the independence numbers of
its powers
Noga Alon
Eyal Lubetzky
June 15, 2005
The independence numbers of powers of graphs have been long studied, under several def-
initions of graph products, and in particular, under the strong graph product. We show that
the series of independence numbers in strong powers of a fixed graph can exhibit a complex
structure, implying that the Shannon Capacity of a graph cannot be approximated (up to a
sub-polynomial factor of the number of vertices) by any arbitrarily large, yet fixed, prefix of the
series. This is true even if this prefix shows a significant increase of the independence number
at a given power, after which it stabilizes for a while.
1 Introduction
Given two graphs, G1 and G2, their strong graph product G1 G2 has a vertex set V (G1) V (G2),
and two distinct vertices (v1, v2) and (u1, u2) are connected iff they are adjacent or equal in each
coordinate (i.e., for i {1, 2}, either vi = ui or viui E(Gi)). This product is associative and
commutative, and we can thus define Gk as the product of k copies of G. In [11], Shannon introduced
the parameter c(G), the Shannon Capacity of a graph G, which is the limit limk


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics