Tales of Hoffman: from a distance

TL;DR

This paper extends Hoffman’s eigenvalue bound for chromatic number to the distance-$k$ setting, introduces a polynomial optimization approach, and explores implications for quantum and vector chromatic numbers.

Contribution

It generalizes Hoffman’s bound to distance-$k$ graphs, introduces a polynomial optimization method, and connects classical and quantum coloring parameters.

Findings

Extended Hoffman bound to distance-$k$ graphs.

Proposed linear programming to optimize polynomial bounds.

Bound is sharp for certain graph classes.

Abstract

Hoffman proved that a graph $G$ with adjacency eigenvalues $\lambda_1\geq \cdots \geq \lambda_n$ and chromatic number $\chi(G)$ satisfies $\chi(G)\geq 1+\kappa,$ where $\kappa$ is the smallest integer such that $$\lambda_1+\sum_{i=1}^{\kappa}\lambda_{n+1-i}\leq 0.$$ We extend this eigenvalue bound to the distance-$k$ setting, and also show a strengthening of it by proving that it also lower bounds the corresponding quantum distance coloring graph parameter. The new bound depends on a degree-$k$ polynomial which can be chosen freely, so one needs to make a good choice of the polynomial to obtain as strong a bound as possible. We thus propose linear programming methods to optimize it. We also investigate the implications of the new bound for the quantum distance chromatic number, showing that it is sharp for some classes of graphs. Finally, we extend the Hoffman bound to the distance setting of the vector chromatic number. Our results extend and unify several previous bounds in the literature.