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.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics