The fasts of Bahab:
The custom: It [used to be] customary to fast a set of three fasts called Bahab after the Holidays of Sukkos and Pesach. [This is no longer the custom amongst most segments of Jewry, including Chabad Chassidim. It is however customary amongst some communities to recite Selichos on these days, although this is not the Chabad custom.] There is no custom to fast Bahab after Shavuos.
When after the holiday do these fasts take place? One only begins the set of fasts after the 1st Shabbos of the next month [i.e. Iyar or Cheshvon], unless that 1st Shabbos falls on Rosh Chodesh, in which case the fast only begin after the 2nd Shabbos of the next month.
 Admur 492:1-5; Michaber 492 in name of Yeish Omrim; Rama 492
 Admur 492:1
 The reason: This was done out of a sign of repentance and atonement as there is suspicion that perhaps due to the festivities of the holiday one was led to transgression. A premise for this is found in Iyov who likewise had sacrifices offered after celebrating with his family out of fear that perhaps his sons sinned. [Admur ibid]
 Hisvadyus 5748 Vol. 3:258
 The reason: As Shavuos is only two days and there is no suspicion that within this time one has sinned. [ibid]
 Admur 492:2 and 4
 The reason: The reason for why we do not fast in the month of Nissan or Tishrei is because the entire month is considered a Holiday and is unbefitting for establishing fatst. [Admur 492:2] Furthermore, the custom is to bless those that are fasting on the Shabbos prior to the fast, and thus being that we do not say such blessings in the months of Tishrei and Nissan, therefore we delay the blessings and the fasts until after the first Shabbos of the next month. Similarly, when Shabbos falls on Rosh Chodesh we do not say such a blessing, and thus in such a case it is pushed off until after 2nd Shabbos of the next month. [Admur 492:4]