For a further introduction to extremal graph theory, see Chapters 7 and 10 of the book Graph Theory by Reinhard Diestel.
The Hajnal-Szemerédi theorem can actually be phrased in the language of equitable colorings rather than clique factors. A short proof of the theorem was given by Kierstead and Kostochka;
see here.
The rotation-extension method (used to prove Dirac's theorem) can give very elegant proofs for Hamilton cycle problems. However, it often is not a viable proof option, particularly in the setting of directed graphs.
In this case the regularity lemma has been widely applied. An example is available in Hamiltonian degree sequences in digraphs.
Lecture 3: an exact result
The notes for this lecture are available here: lecture notes.
Lecture 4: the Lo-Markström lemma
The slide for this lecture is available here: Lecture 4 slide.
The Lo-Markström lemma presented is a special case of Lemma 1.1 from F-factors in hypergraphs via absorption (which is stated more generally about hypergraphs).
Some further applications of this lemma can be found in this paper also.
The statement of the Balogh-Kostochka-Treglown conjecture extends to clique factors more generally; see Conjecture 7 in On perfect packings in dense graphs.
The slide for this lecture is available here: Lecture 5 slide.
For some more background on the regularity lemma see Chapters 7 and 9 of the book Graph Theory by Reinhard Diestel. In particular, the regularity lemma is also useful in situations
other than embedding (almost) spanning subgraphs. An appication in Ramsey theory can be found in Chapter 9.
Further background on epsilon-regular pairs and the regularity lemma can be found in an old thesis of mine.
The blow-up lemma presented in class is a special case of that presented in the paper
Blow-up Lemma
(see also, e.g., the survey
The Blow-up Lemma).
Lecture 7: Montgomery's method
The notes for this lecture are available here: lecture notes.