The annual Torah reading cycle has us reading the story of Yosef during Chanukah. While in most years it is Parshat Mikketz we read on Shabbat Chanukah, the Yosef story begins in Parshat Vayeshev which in approximately 10% of years is also Shabbat Chanukah. Of course, the linkage of Yosef and Chanukah is actually a coincidence having nothing to do with Yosef or Chanukah. If one is going to begin the reading of Parshat Breisheet on the shabbat after Simchat Torah, when Chanukah arrives some two months later, we will, by default, be reading Vayeshev or Mikketz.
This is true, however, only according to the Babylonian a cycle of reading the Torah – one that divided the Torah into 54 sidrot and is now the standard practice the world over. But according to the ancient practice in Israel, in which it took over three years to complete the Torah reading, the reading on Shabbat Chanukah would vary from year to year.
The Gemara does list a number of Torah portions that are to be read at specific times – for example, reading the tocheicha, the admonitions to ensure we are faithful to the Torah way of life, both before Shavuot and Rosh Hashanah – but Yosef and Chanukah is not amongst them. The Mishna (Megillah 30b) does instruct us to read the Torah on Chanukah, but it is parshat nesi'im (Bamidbar chapter 7), the gifts brought by the leaders of the 12 tribes to the original dedication of the Mishkan in the desert, that we are to read. And that is what we read each of the eight days of Chanukah, serving as the maftir portion for Shabbat Chanukah. Yosef and Chanukah are apparently no more than a historical fluke.
Nonetheless, coincidence or not, there is much that links Yosef and Chanukah. Both deal with the issue of maintaining Jewish identity in a foreign culture. This is the primary question that Jews have been struggling with in the modern period (and likely beforehand too) – and this struggle has led to such iterations of Judaism as Reform, Zionism – secular and religious, Modern Orthodoxy, Rejectionist (of modernity) Orthodoxy, and more.
In order for the Jewish people to become a great nation we were going to have to spend 400 years in a foreign land. This exposure to another culture is so fundamental to our essence that it is the basis of the Brit bein Habetarim, the covenant G-d made with Abraham. And it is for good reason that G-d choose Egypt. It was the most advanced country of the day - technologically, economically, politically and in many more ways. We still do not quite know how they managed to build the pyramids. The Jewish people were to learn how to be the best, world leaders, from their host country. Our advanced knowledge would sanctify the name of G-d, impressing the nations of the world who would proclaim, "that great nation [Israel] is a wise and discerning people" (Devarim 4:6).
At the same time Egypt was morally corrupt – the Torah specifically warns us not to follow their sexual (Vayikra, chapter 18) and business practices (Vayikra 19:35-36). And what is perhaps the overriding message of the Torah – it is by far the most repeated one - is that unlike the Egyptians we are to treat the stranger with extra sensitivity.
Separating the good from the bad is no easy task – nothing meaningful is, or should be, easy – but that is the task of the Jewish people. Yosef nearly failed in this task and in the understanding of the Sages came so very close to - like Lot, Yishmael and Eisav before him - being lost to the Jewish people. Yet Yosef, despite changing his name, language and manner of dress, and despite his desire to forget his past (just look at the names he choose for his children) when push came to shove revealed his true self, “I am your brother Yosef” (Breisheet 45:4).
It is not by chance that we bless our sons to be like Ephraim and Menashe. They were the first Jewish children to be born, raised and live in a foreign environment and yet remain faithful to their ancient heritage.
Chanukah commemorates the victory of the Maccabees over the Hellenists. Fascinatingly, of all the commentators, it is the Rambam who most emphasizes the defeat of the Greeks, relegating the miracle of the oil to a secondary role (see the Laws of Megillah and Chanukah, 3:1-2). Yet this same Rambam embraced much of Greek culture – and for good reason. The ancient Greeks laid the foundations of philosophy, democratic politics, logic, mathematics, medicine, architecture and so much more – benefitting the world in so many ways. Yet the Greeks worshipped the body, believed in human gods, created many a myth and attempted to eradicate fundamental Jewish practices.
There is much that is bad but much that is good in ancient Greek culture. And it is our task to distinguish between the two, rejecting one and embracing the other. As our Sages teach, the essence of wisdom is to make distinctions.
It may be a coincidence that we read the story of Yosef on Chanukah but what a coincidence it is.
 At the same time, the parallels between the Yosef story and Megillat Esther are such that is clear the author of the Megillah had the Yosef story in front of him (them?) as they wrote the Megillah. Both stories, to cite some examples, revolve around young Jews, who against all odds rise to power in the most powerful empire of the day. Their rise to power is a result of their good looks and while both come very close to abandoning their Jewishness, they use their power to save the Jewish people. In both, the story takes a major positive turn due to the King’s insomnia. The parallels in language are many and striking. For a comprehensive analysis of the link between Yosef and Megillat Esther please see the introduction to the Megillah in the Da'at Mikra series of Mossad Harav Kook.
 The only calendric configuration when Yayeshev is shabbat Chanukah is when Shmini Atzeret is on Shabbat and the months of Cheshvan and Kislev (the only months which can be either 29 or 30 days) are both 29 days.
 As he was about to sin with Potiphar's wife he somehow managed to see the image of his father and that gave him the strength to resist and not be lost to our people (see Rashi, Breisheet 39:11)
 This is why we say havdalah in the bracha of chonen hada'at where we ask G-d to grant us wisdom.