On the Normalized Shannon Capacity of a Union

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  • University of Oxford


Let G1 × G2 denote the strong product of graphs G1 and G2, that is, the graph on V(G1) × V(G2) in which (u1, u2) and (v1, v2) are adjacent if for each i = 1, 2 we have ui = vi or uivi ∈ E(Gi). The Shannon capacity of G is c(G) = limn → ∞ α(Gn)1/n , where Gn denotes the n-fold strong power of G, and α(H) denotes the independence number of a graph H. The normalized Shannon capacity of G is

C(G) = log c(G) / log |V(G)|.

Alon [1] asked whether for every ε < 0 there are graphs G and G′ satisfying C(G), C(G′) < ε but with C(G + G′) > 1 − ε. We show that the answer is no.


Original languageEnglish
Pages (from-to)766-767
Number of pages2
JournalCombinatorics, Probability and Computing
Issue number5
Early online date3 Mar 2016
Publication statusPublished - 1 Sep 2016


  • Shannon capacity