Hoffman colorability of (strongly) regular graphs
Abstract
Hoffman's bound is a well-known eigenvalue bound on the chromatic number of a graph. By interpreting this bound as a parameter, we show multiple applications of colorings attaining the bound (Hoffman colorings) for several notions of graph regularity: regular, (co-)edge-regular, and strongly regular. For strongly regular graphs, we prove that Hoffman colorability implies pseudo-geometricity, and we strengthen Haemers' finiteness result on strongly regular graphs with a bounded chromatic number by considering the Hoffman bound instead of the chromatic number. Furthermore, by using Hoffman colorings we show that a sufficient condition for non-unique vector colorability shown by Godsil, Roberson, Rooney, \v{S}\'amal and Varvitsiotis [European J. Combin. 79, 2019] can be relaxed in the setting of strongly regular graphs. Lastly, using Hoffman colorings we derive several new characterizations of the mentioned graph regularity notions.
Cite
@article{arxiv.2508.18793,
title = {Hoffman colorability of (strongly) regular graphs},
author = {Aida Abiad and Bart De Bruyn and Thijs van Veluw},
journal= {arXiv preprint arXiv:2508.18793},
year = {2025}
}