One simple remark is that you use the induction hypothesis where one person knows two rumors instead of one. The induction hypothesis and the base case need to be relaxed so that it is not essential how many rumors each person knows in the beginning.

More importantly, by providing a way to exchange rumors in 2n - 4 calls, you only proved that G(n), theminimumnumber of calls, is <= 2n - 4. For the converse it is probably essential that the sets of rumors each person knows are pairwise disjoint.