# Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

## TL;DR

This paper extends the computation of the graph complexity invariant for orientable 3-manifolds with toric boundary, providing exact values for an infinite family of tetrahedral manifolds and bounds for others.

## Contribution

It advances the calculation of graph complexity for a broader class of 3-manifolds, including new exact values and bounds for tetrahedral manifolds.

## Key findings

- Exact graph complexity computed for an infinite family of tetrahedral manifolds.
- Extended graph complexity calculations to manifolds with toric boundary up to complexity 14.
- Provided two-sided bounds for the graph complexity of tetrahedral manifolds.

## Abstract

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequence, we compute the exact value of this invariant for an infinite family of such manifolds.