Characterizing when the Nikiforov bound improves on Hoffman's bound

Determine a simple characterization of the bounded-degree Borel pmp graphs, particularly non-regular graphs, for which the approximate measurable Nikiforov spectral lower bound on the chromatic number is tighter than the corresponding approximate measurable Hoffman bound.

Background

The paper establishes an approximate measurable analogue of Nikiforov’s chromatic-number bound involving the maximum spectral values of the adjacency and Laplacian operators. In the regular case, this bound reduces to the Hoffman bound, whereas for non-regular graphs the two bounds can be incomparable, as in the classical finite-graph setting.

The authors identify the absence of a simple criterion distinguishing which graphs benefit from the Nikiforov bound as an unresolved problem. A solution would explain structurally when the new non-regular spectral estimate gives genuinely stronger information than the Hoffman estimate.