Hoffman colorings of graphs

TL;DR

This paper explores the structure and properties of Hoffman colorings in graphs, especially irregular ones, providing a decomposition theorem, classifications, and algorithms to understand Hoffman colorability and related coloring parameters.

Contribution

It introduces a Decomposition Theorem for Hoffman colorings, classifies Hoffman colorability in cone and line graphs, and develops an algorithm for identifying Hoffman colorable graphs.

Findings

Decomposition Theorem characterizes Hoffman colorings.

Complete classification for cone and line graphs.

Algorithm for computing Hoffman colorable graphs.

Abstract

Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.