Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-15T10:23:48.271Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Normalized Shannon Capacity of a Union

Part of: Communication, information Graph theory

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alon, N. (1998) The Shannon capacity of a union. Combinatorica 18 301–310.Google Scholar
[2] Alon, N. (2002) Graph powers, Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 11–28.Google Scholar
[3] Alon, N. and Lubetzky, E. (2006) The Shannon capacity of a graph and the independence numbers of its powers. IEEE Trans. Inform. Theory 52 2172–2176.CrossRefGoogle Scholar
[4] Alon, N. and Orlitsky, A. (1995) Repeated communication and Ramsey graphs. IEEE Trans. Inform. Theory 41 1276–1289.CrossRefGoogle Scholar
[5] Haemers, W. (1979) On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 231–232.CrossRefGoogle Scholar
[6] Körner, J. and Orlitsky, A. (1998) Zero-error information theory. IEEE Trans. Inform. Theory 44 2207–2229.CrossRefGoogle Scholar
[7] Lovász, L. (1979) On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 1–7.CrossRefGoogle Scholar
[8] Shannon, C. E. (1956) The zero-error capacity of a noisy channel IRE Trans. Inform. Theory 2 8–19.CrossRefGoogle Scholar