Efficient spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers
Finn A. Steinke, Luis M. B. Varona
TL;DR
This work addresses the problem of obtaining computable lower bounds on the chromatic number χ for powers of structured graphs: Hamming, Johnson, and Kneser graphs. It develops a spectral approach based on the Hoffman bound and leverages association schemes to derive closed-form eigenvalues in terms of Kravchuk and Eberlein polynomials, enabling dynamic-programming algorithms that run in O($np$), O($kp$), and O($k^2$) time for H(n, q)^p, J(n, k)^p, and K(n, k)^p, respectively. The key contributions are the first nontrivial, scalable chromatic-lower-bound computations for these graph powers, with results that generalize beyond hypercube powers and scale polynomially in the relevant parameters. These methods have practical implications for coding theory, error correction, and distributed computing, and open avenues for extending to other spectral bounds and vertex-transitive graph families.
Abstract
We investigate spectral lower bounds on the chromatic number $χ$ 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 $χ$ can, in principle, be applied to any graph, naïve 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.