An inertial lower bound for the chromatic number of a graph

TL;DR

This paper introduces an inertial lower bound for the chromatic number of a graph based on its inertia, explores extremal cases, and discusses related bounds and conjectures in graph coloring theory.

Contribution

It establishes a new inertial lower bound for the chromatic number and investigates its properties and limitations compared to other coloring parameters.

Findings

Inertial bound provides a new lower bound for the chromatic number.

The inertial bound is not valid for the vector chromatic number.

Discussion includes extremal graphs and Nordhaus-Gaddum bounds for inertia.

Abstract

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that: \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{ and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G) \] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.