“….In the tenth month, on the tenth of the month… the king of babylon has besieged jerusalem on this very day….” – Yechezkel 24:1-2

״….בחודש העשירי בעשור לחודש…. סימך מלך בבל אל ירושלים בעצם היום הזה…״ – יחזקאל כ״ד, א-ב

The early and important work known as Sefer Abudraham states that if the 10^thof Teves were to occur on a Shabbos we would not postpone the fast, (as we do when all other fast days besides Yom Kippur fall out on Shabbos.) He bases this on the prophet Yechezkel’s reference to this day with the words, ‘בעצם היום הזה – on this very day,’ the same expression used by the Torah when speaking of the fast day of Yom Kippur, which is observed even when it occurs on Shabbos.

The Halacha is not ruled in accordance with this opinion, nor is it possible for the 10^th of Teves to occur on Shabbos according to our current calendar. (Nevertheless, we do fast until nightfall on the 10^th of Teves even when it occurs on a Friday, thus fasting into Shabbos, see Likkutei Sichos vol. 25, p. 267, fn. 5.) This opinion highlights that the Fast of the 10^th of Teves is more severe than the other minor fasts, as the calamity which occurred on this day was, in a sense, the father of the later calamities which are observed with the other minor fast days. The Fast of the 10^th of Teves commemorates the siege on Yerushalayim, which led to  the series of calamities that followed: the breach of the wall, the destruction of the Beis Hamikdash and the exile and dispersion of the Jewish nation.

This explains the comparison of this fast to Yom Kippur, both referred to with the phrase “on this very day.” For the harshness of the tragedy which occurred on this day is likewise reflected in the feelings of teshuva that this fast evokes, which are the most profound of all the minor fast days, regretting and repairing the mistakes that brought about this “father of all tragedies.” Thus, this teshuvais even comparable to the teshuva of Yom Kippur, which is observed by fasting even if it occurs on Shabbos.

Furthermore, just as the restraint from food on Yom Kippur is comparable to the pleasure of Olam Haba, the World to Come, of which our Sages say “there is no eating and drinking (Brachos 17a),” so are the effects of the teshuvaachievable on the 10th of Teves. Their delight transcends even the pleasure of eating on Shabbos.

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—Likkutei Sichos, vol. 15, pp. 420-421