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On the Normalized Shannon Capacity of a Union

Part of: Graph theory Communication, information

Published online by Cambridge University Press:  03 March 2016

PETER KEEVASH  and
EOIN LONG
PETER KEEVASH
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, UK (e-mail: keevash@maths.ox.ac.uk)
EOIN LONG
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: eoinlong@post.tau.ac.il)
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Abstract

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

$$C(G) = \ffrac {\log c(G)}{\log |V(G)|}.$$
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.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 94A05: Communication theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 5 , September 2016 , pp. 766 - 767
Copyright
Copyright © Cambridge University Press 2016 

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References

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