The Shannon Capacity of a Union

For an undirected graph G = (V, E), let Gn denote the graph whose vertex set is V n 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.

• Type: Journal
• Year: 1998
• Where: COMBINATORICA
• Authors: Noga Alon
• Keywords: Adjacent Iff, COMBINATORICA 1998, Distinct Vertices, Undirected Graph