  The pants graph has proved to be influential in understanding 3-manifolds
concretely. This stems from a quasi-isometry between the pants graph and the
Teichm\"uller space with the Weil-Petersson metric. Currently, all estimates on
the quasi-isometry constants are dependent on the surface in an undiscovered
way. This paper starts effectivising some constants which begins the
understanding how relevant constants change based on the surface. We do this by
studying the hyperbolicity constant of the pants graph for the five-punctured
sphere and the twice punctured torus. The hyperbolicity constant of the
relative pants graph for complexity 3 surfaces is also calculated. Note, for
higher complexity surfaces, the pants graph is not hyperbolic or even strongly
relatively hyperbolic.
