  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.
