On the Normalized Shannon Capacity of a Union

Despite much impressive work (e.g. [1], [3], [4], [5], [7]) since the introduction of the Shannon capacity in [8], many natural questions regarding this parameter remain widely open (see [2], [6] for surveys). Let G1 + G2 denote the disjoint union of the graphs G1 and G2. It is easy to see that c(G1 + G2) ≥ c(G1) + c(G2). Shannon [8] conjectured that c(G1 + G2) = c(G1) + c(G2), but this was disproved in a strong form by Alon [1] who showed that there are n-vertex graphs G1, G2 with c(Gi) < e c √ logn log logn but c(G1 + G2) ≥ √ n. In terms of the normalized Shannon capacity, this implies that for any > 0, there exist graphs G1, G2 with C(Gi) < but C(G1 +G2) > 1/2− . Alon [1] asked whether ‘1/2’ can be changed to ‘1’ here. In this short note we will give a negative answer to this question. In fact, the following result implies that ‘1/2’ is tight.