On the chromatic number of a simplicial complex

TL;DR

This paper extends Hoffman's eigenvalue-based lower bound on graph chromatic number to higher-dimensional simplicial complexes using spectra of higher Laplacian operators.

Contribution

It introduces a novel spectral bound for the chromatic number of pure simplicial complexes, generalizing Hoffman's graph result.

Findings

Provides a spectral lower bound on the chromatic number of simplicial complexes

Generalizes Hoffman's bound from graphs to higher dimensions

Connects spectral properties of Laplacians with coloring properties

Abstract

In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a lower bound on the chromatic number of a pure $d$-dimensional simplicial complex in the terms of the spectra of the higher Laplacian operators.