• algebraic combinatorics
• chromatic quasisymmetric functions
• interval graphs
• Shareshian-Wachs conjecture
• Stanley-Stembridge conjecture

The purpose of this thesis is to deal with chromatic quasisymmetric functions and the Shareshian-Wachs conjecture.
In the first part, we gradually introduce the problem, showing which are the main reasons why it is studied.
We begin with graph colorings, defining quasisymmetric chromatic functions and introducing the Stanley-Stembridge conjecture, concerning the positivity of the coefficients of these functions in the expansion in the elementary basis.
Then we show a proof by Guay-Paquet in which he reduces the hypothesis of this conjecture to a more restricted class of graphs.
At this point, adding an additional statistic on the colorings, we generalize the functions to chromatic quasisymmetric functions and state the Shareshian-Wachs conjecture.
Finally, we see some connections that the latter conjecture has with LLT polynomials and Hessenberg varieties.
In the second part we do a review of the existing literature, focusing on the expansion of some particular chromatic quasisymmetric functions in the main bases of symmetric and quasisymmetric functions.
In a first chapter we show relationships between existing formulae, which use structures such as sets of permutations and acyclic orientations, and we also introduce a new structure called "magic spanning forest".
In a second chapter we analyze another approach to the problem by Abreu and Nigro that uses the so-called "modular law" and try to understand the difficulties in generalizing this method to a larger class of graphs.