pants graph is strongly relatively hyperbolic, specifically when $\xi(S) = 3$ [@BM]. Even though hyperbolicity is well studied for the pants graph, the hyperbolicity constants associated with the pants graph or the relative pants graph is not. In addition to having a further understanding of the quasi-isometry mentioned above and all of its applications, actual hyperbolicity constants are useful in answering questions about asymptotic time complexity of certain algorithms, especially those involving the mapping class group. More speculatively, estimates on hyperbolicity constants may be crucial to effectively understand the virtual fibering conjecture, which relates the geometry of the fiber to the geometry of the base surface. The focus of this paper is to find hyperbolicity constants for the pants graph and relative pants graph, when these graphs are hyperbolic.

For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.

Computing the asymptotic translation lengths of an element in the mapping class group on ${\mathcal{P}}(S)$ is a question explored by Irmer [@Irmer]. Bell and Webb have an algorithm that answers this question for the curve graph [@BellWebb]. Combining the works of Irmer, and Bell and Webb, one could conceivably come up with an algorithm for asymptotic translation lengths on ${\mathcal{P}}(S)$. 