On the Normalized Shannon Capacity of a Union

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

In: Combinatorics, Probability and Computing, Vol. 25, No. 5, 01.09.2016, p. 766-767.

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@article{34a0f5daad004bf6852dabd90cac2dd0,
title = "On the Normalized Shannon Capacity of a Union",
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 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.",
keywords = "Shannon capacity",
author = "Peter Keevash and Eoin Long",
year = "2016",
month = sep,
day = "1",
doi = "10.1017/S0963548316000055",
language = "English",
volume = "25",
pages = "766--767",
journal = "Combinatorics, Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "5",

}

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 -