This paper proposes an adaptation of the TCC mechanism. We have seen that the TCC mechanism delivers the Walrasian allocation but can also have undesirable consequences, as is the case in Boston school choice, where it was rejected in favor of the DA algorithm.
This paper attempts to bring equity to the TTC, they call it the Equitable Top Trading Cycles (ETTC). The TCC mechanism is both Pareto efficient and strategy-proof. It can, however, bring equity concerns. They propose the following mechanism: Each student-school pair points to the student-school pair that contains the highest priority student for the school contained in the initial pair. The mechanism is described in section 4.1, there is an example that is particularly helpful to understand). This mechanism is both Pareto efficient and strategy-proof.
It is an interesting paper not only because it introduce an equitable mechanism, but also because it mentions many potential shortcomings of TCC in the context of school assignment. Also, it is helpful to read section 3 for an easy description of the TCC mechanism.