Hoffman colorability of graphs with smallest eigenvalue at least -2

TL;DR

This paper extends the classification of Hoffman colorability to all connected graphs with smallest eigenvalue at least -2, including line graphs, generalized line graphs, and exceptional graphs, with detailed classifications and maximality results.

Contribution

It provides a comprehensive characterization of Hoffman colorability for graphs with smallest eigenvalue at least -2, expanding previous results to all connected graphs and classifying exceptional cases.

Findings

245 Hoffman colorable exceptional graphs identified

29 maximal Hoffman colorable graphs classified

39 graphs maximal in E7 root system representation

Abstract

In accordance with the Cameron-Goethals-Seidel-Shult Classification Theorem, we extend the characterization of Hoffman colorability of line graphs from (Abiad, Bosma, Van Veluw, 2025) to all connected graphs with smallest eigenvalue at least $-2$; we give a characterization of Hoffman colorability of generalized line graphs, and we completely classify the Hoffman colorable exceptional graphs. The 245 Hoffman colorable exceptional graphs from this classification admit a natural partial ordering, and we determine the 29 graphs that are maximal in this respect, in a way similar to the classification of maximal ($E_8$-representable) exceptional graphs as described in (Cvetkovi\'c, Rowlinson, Simi\'c, 2004). Lastly, as a byproduct and also similarly as in (loc. cit.), we determine all 39 graphs that are maximal with respect to being representable in the $E_7$ root system.