# On the Normalized Shannon Capacity of a Union

Research output: Contribution to journal › Article › peer-review

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**On the Normalized Shannon Capacity of a Union.** / Keevash, Peter; Long, Eoin.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Combinatorics, Probability and Computing*, vol. 25, no. 5, pp. 766-767. https://doi.org/10.1017/S0963548316000055

## APA

*Combinatorics, Probability and Computing*,

*25*(5), 766-767. https://doi.org/10.1017/S0963548316000055

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## Bibtex

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## RIS

TY - JOUR

T1 - On the Normalized Shannon Capacity of a Union

AU - Keevash, Peter

AU - Long, Eoin

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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 isC(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.

AB - 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 isC(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.

KW - Shannon capacity

U2 - 10.1017/S0963548316000055

DO - 10.1017/S0963548316000055

M3 - Article

VL - 25

SP - 766

EP - 767

JO - Combinatorics, Probability and Computing

JF - Combinatorics, Probability and Computing

SN - 0963-5483

IS - 5

ER -