What connections does Chanuka have with Shavuos?
Looking at book: Divine Design by Moshe Shlomo Emanuel (here).
He lists various connections (summarized here):
The connetions between Chanuka and Shavuos include those answered, which in summary are:
A: Eating dairy on both.
B: Both celebrate the Torah.
C: Both connect to the number 8.
In addition to the above there are also the following connections.
A: Both Chanuka and Shavuos can technically come out on different days of the month. Shavuos is 50 days after Pesach. Although in the current calendar it always comes out on 6 Sivan, in the times of Kiddush HaChodesh it could come out as early as 5 Sivan and as late as 7 Sivan. (That is why we say "Zeman" Matan Toroseinu - instead of "Yom" Matan Toroseinu). Chanuka can end on either 2 Teves or 3 Teves.
B: According to the Shulchan Aruch Siman 428 the day of the week Chanuka starts is the same day Shavuos will start. The Mishna Berura says that it is an error and it is reversed to the day of the week Shavuos starts that will be the same day of the week the following Chanuka will start. However the Mishna Berura mentions that it only works if Cheshvan is 29 days, however if Cheshvan is 30 days then Chanuka will come out one day later in the week than Shavuos.