Hoffman colorability of graphs with smallest eigenvalue at least -2

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 (Cvetković, Rowlinson, Simić, 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.

Figures (10)

• Figure 1: The smallest connected non-trivially Hoffman colorable graph.
• Figure 2: A sporadic Hoffman edge coloring.
• Figure 3: Smith graphs $\mathcal{F}_7, \mathcal{F}_8, \mathcal{F}_9$, with eigenvectors for the largest eigenvalue 2.
• Figure 4: Edge switching diagrams for the three Chang graphs and the Schläfli graph.
• Figure 5: Edge switching diagram for $M_{20}$.

Theorems & Definitions (62)

• Theorem 2.1
• Theorem 2.2
• Proposition 3.1: spectra
• Theorem 3.2: ratio bound, Hoffman (unpublished, see ratiobound)
• Proposition 3.3: 3chromDRG
• Theorem 3.4: Decomposition Theorem, previouspaper
• Theorem 3.5: previouspaper
• Remark 3.6
• Theorem 3.7: previouspaper
• Proposition 3.8: previouspaper