# More Tales of Hoffman: bounds for the vector chromatic number of a graph

## TL;DR

This paper extends spectral lower bounds from the chromatic number to the vector chromatic number, providing new bounds and a characterization that deepen understanding of graph coloring complexities.

## Contribution

It proves that additional spectral lower bounds for the chromatic number also apply to the vector chromatic number, and introduces a new characterization of the latter.

## Key findings

- Spectral bounds for $ ext{chi}(G)$ are valid lower bounds for $ ext{chi}_v(G)$.
- Two new spectral lower bounds for $ ext{chi}(G)$ are established for $ ext{chi}_v(G)$.
- A new characterization of the vector chromatic number is derived using these bounds.

## Abstract

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower bound for $\chi(G)$ is in fact a lower bound for $\chi_v(G)$. We prove that two more spectral lower bounds for $\chi(G)$ are also lower bounds for $\chi_v(G)$. We then use one of these bounds to derive a new characterization of $\chi_v(G)$.