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We study equality in the Hoffman bound for the chromatic number and Hoffman colorings in regular and irregular graphs. We investigate the connection between Hoffman colorability and several graph operations, of which the tensor product is especially interesting in this context. We then introduce the Decomposition Theorem revealing structural properties that Hoffman colorings must obey. Using the Decomposition Theorem we are able to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, allowing us to find various new infinite families of Hoffman colorable graphs, many of which are irregular. Lastly we introduce a new parameterization and type system for strongly regular graphs, that show connections between Hoffman colorability, spreadability, pseudo-geometricity and unique vector colorability.
Thijs van Veluw (Tue,) studied this question.