TY - JOUR
T1 - The Shannon capacity of a union
AU - Alon, Noga
N1 - Funding Information:
Mathematics Subject Classi cation (1991): 05C35, 05D10, 94C15 Part of this work was done at DIMACS and at the Institute for Advanced Study, Princeton, NJ 08540, USA. Research supported in part by a grant from the the Israel Science Foundation, by a State of New Jersey grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.
PY - 1998
Y1 - 1998
N2 - For an undirected graph G = (V, E), let Gn denote the graph whose vertex set is Vn in which two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent iff for all i between 1 and n either ui = vi or uivi ∈ E. The Shannon capacity c(G) of G is the limit limn→∞ (α(Gn))1/n, where α(Gn) is the maximum size of an independent set of vertices in Gn. We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956.
AB - For an undirected graph G = (V, E), let Gn denote the graph whose vertex set is Vn in which two distinct vertices (u1, u2, . . . , un) and (v1, v2, . . . , vn) are adjacent iff for all i between 1 and n either ui = vi or uivi ∈ E. The Shannon capacity c(G) of G is the limit limn→∞ (α(Gn))1/n, where α(Gn) is the maximum size of an independent set of vertices in Gn. We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956.
UR - http://www.scopus.com/inward/record.url?scp=0032411859&partnerID=8YFLogxK
U2 - 10.1007/PL00009824
DO - 10.1007/PL00009824
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AN - SCOPUS:0032411859
VL - 18
SP - 301
EP - 310
JO - Combinatorica
JF - Combinatorica
SN - 0209-9683
IS - 3
ER -