---
abstract: '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 Teichmüller 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.'
author:
- Ashley Weber
bibliography:
- 'mybib.bib'
date:  
title: Hyperbolicity constants for pants and relative pants graphs
---

Introduction
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The pants graph has been instrumental in understanding Teichmüller space. This is because the pants graph is quasi-isometric to Teichmüller space equipped with the Weil-Petersson metric [@Brock-WPtoPants]. Brock and Margalit used pants graphs to show that all isometries of Teichmüller space with the Weil-Petersson metric arise from the mapping class group of the surface [@BM-WPisom]. This relationship was also used to classify for 