8th of Tevet: the event with Ptolemy who forced the 70 chachamim to translate the Torah
9th of Tevet: Ezra and Nechemia had died.
10th of Tevet: Siege around Yerushalayim
The tenth of Tevet is considered such a significant day for fasting and teshuva that even if it were to fall out on Shabbat, one would not push off the fast and the fast day would take place on Shabbat. Since the establishment of the calendar, the 10th of Tevet can not occur on Shabbat.
The Beit Yosef in Orach Chaim, 550, quoting the Abudaraham, mentions that were Asara B'Tevet to fall out on Shabbat, one would fast. The reasoning is due to the actual date that is mentioned in Ezekiel: 24:1-2. It also appears in Jeremiah 52:6-7.
The Shulchan Aruch's final ruling, however, is that, as with all the 4 minor fasts, one would not fast on Shabbat.
However, when the 10th of Tevet occurs on a on Friday, the 10th of Tevet is not postponed. Some have made this the general Memorial Day for those who perished during the Shoah and Kaddish is recited.
May the 10th of Tevet be a day involving
Teshuva, Tefilah and Tzedakah