Reading on a fast without 10 fasting

A question came to the attention of my Minyan in Chicago this past fast-day (the 10th of Teveth) and we would appreciate your opinion, with sources, for the future. There were present at Mincha prayers only three men who were fasting. Would it have been Halachicly possible to have read from the Tora portion in such a situation? What is the Halacha and the opinion of the Rabbanim (with their sources). Thank you for your advice.

It is not possible to read from the Torah on a fast day even if the fast is obligatory one i.e. one of the four fast days mentioned in the Pasuk, unless there are at least seven fasters (Mishna Berura OC 566,14) quoting the Sha'are Teshuva 4 in the name of Maharam Ben Chaviv). If it is a volantary fast one needs all 10 in the minyan to be fasters (Mishna Berura). This ruling is based on the Tur (566,3) in the name of his father the Rosh that we need a Minyan of fasters in order that the gathering be considered a 'congregation' to enact the special prayers. On a normal day Torah reading the Chayai Adam was undecided if one could reread with less than a Minyan who have not yet heard the reading. The Mishna Berura rules (OC 143,1 Beur Halacha 'Bepachot') that one can reread the Parasha for a majority of a Minyan that have not yet heard the reading. If there is less than a majority one cannot reread the portion. From this we see that Torah reading is stricter than other parts of the prayers and for a fast day you need at least a majority who are fasting to read the Torah. On a fast day and a Thursday morning one can read the special reading with only 6 people are fasting as opposed to the need of 7 men for any other reading.(Mishna Berura ibid). Rabbi Baruch Paz