Lectures 4 & 5 : Shannon Capacity of a Union of Graphs

In the previous lectures, we have defined the Shannon Capacity of a graph, Θ(G). This lecture is concerned with the Shannon Capacity of the disjoint union G +H of two graphs G and H. The union of two graphs is, informally, what you get when you just place one graph next to the other. In 1956, Shannon conjectured [1] that Θ(G + H) = Θ(G) + Θ(H). As we will see, this is a very natural conjecture that assumes that when two completely disjoint alphabets are used for zero-error communication, there is no benefit in jointly coding over the two alphabets. Surprisingly, it was disproved 42 years later by N. Alon [3]. We will present the simpler construction from [4]. The requirement is the construction of a counterexample: two graphs G and H such that the Shannon capacity Θ(G+H) of their disjoint union is strictly bigger than the sum of their capacities, i.e., Θ(G+H) > Θ(G) + Θ(H). In addition to a specific construction, we develop a technique to upper bound the sizes of independent sets. In the previous lecture we saw that the Lovasz function is an efficiently computable bound on the sizes of independent sets of a graph and its powers. Here we will see how a dimension argument can be used to obtain a different bound on independent sets. One key property of the proof is the use of dimension arguments over two different fields, one of even and one of odd characteristic.