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On the Normalized Shannon Capacity of a Union

  • PETER KEEVASH (a1) and EOIN LONG (a2)

Abstract

Let G 1 × G 2 denote the strong product of graphs G 1 and G 2, that is, the graph on V(G 1) × V(G 2) in which (u 1, u 2) and (v 1, v 2) are adjacent if for each i = 1, 2 we have ui = vi or u i v iE(G i ). 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.

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References

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[1] Alon, N. (1998) The Shannon capacity of a union. Combinatorica 18 301310.
[2] Alon, N. (2002) Graph powers, Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 11–28.
[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 21722176.
[4] Alon, N. and Orlitsky, A. (1995) Repeated communication and Ramsey graphs. IEEE Trans. Inform. Theory 41 12761289.
[5] Haemers, W. (1979) On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 231232.
[6] Körner, J. and Orlitsky, A. (1998) Zero-error information theory. IEEE Trans. Inform. Theory 44 22072229.
[7] Lovász, L. (1979) On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 17.
[8] Shannon, C. E. (1956) The zero-error capacity of a noisy channel IRE Trans. Inform. Theory 2 819.

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On the Normalized Shannon Capacity of a Union

  • PETER KEEVASH (a1) and EOIN LONG (a2)

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