Efficient spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers

TL;DR

This paper develops efficient spectral bounds for the chromatic number of certain graph powers, enabling feasible computation of these bounds using eigenvalues expressed via orthogonal polynomials.

Contribution

It introduces a method to compute spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers efficiently through eigenvalue analysis and recurrence relations.

Findings

Spectral bounds can be computed in polynomial time for these graph classes.

Eigenvalues are expressed in terms of hypergeometric orthogonal polynomials.

Dynamic programming enables efficient computation of Hoffman bounds.

Abstract

We investigate spectral lower bounds on the chromatic number $\chi$ of Hamming graph powers $H(n, q)^p$, Johnson graph powers $J(n, k)^p$, and Kneser graph powers $K(n, k)^p$ providing the first computationally feasible nontrivial results. While the classical Hoffman bound on $\chi$ can, in principle, be applied to any graph, na\"ive computation requires $O(q^{3n})$ time for $H(n, q)^p$ and $O(({}_nC_k)^3)$ time for both $J(n, k)^p$ and $K(n, k)^p$. We thus express the adjacency eigenvalues of these graphs in terms of hypergeometric orthogonal polynomials, exploiting recurrence relations that arise to efficiently compute the entire spectra. We then apply dynamic programming to compute the Hoffman bounds for $H(n, q)^p$, $J(n, k)^p$, and $K(n, k)^p$ in $O(np)$, $O(kp)$, and $O(k^2)$ time, respectively.