Hoffman colorability of (strongly) regular graphs

TL;DR

This paper explores Hoffman colorings in regular graphs, revealing new links between eigenvalue bounds, graph regularity, and colorability, and providing strengthened results and characterizations for strongly regular graphs.

Contribution

It introduces the concept of Hoffman colorability for various regular graphs, establishes its implications for strongly regular graphs, and offers new characterizations and relaxed conditions for vector colorability.

Findings

Hoffman colorability implies pseudo-geometricity in strongly regular graphs.

Strengthened finiteness results for strongly regular graphs with bounded chromatic number.

New characterizations of regularity notions using Hoffman colorings.

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.