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Hoffman colorings of graphs

Aida Abiad, Wieb Bosma, Thijs van Veluw

TL;DR

This work studies when Hoffman's spectral bound on the chromatic number, $h(G)=1-\frac{\lambda_{\max}(G)}{\lambda_{\min}(G)}$, is tight for irregular graphs. It develops the Decomposition Theorem, which imposes a weight-regular structure on Hoffman colorings with at least three colors, and the Composition Theorem, enabling construction of larger Hoffman colorable graphs from templates. These tools yield complete classifications for the Hoffman colorability of cone graphs and line graphs, and underpin an algorithm to enumerate all connected Hoffman colorable graphs with a given $n$ and $\chi$, with empirical results up to six colors. As a byproduct, the authors determine values of several coloring parameters lying between Hoffman's bound and the chromatic number, and provide extensive computational data illustrating the landscape of 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.

Hoffman colorings of graphs

TL;DR

This work studies when Hoffman's spectral bound on the chromatic number, , is tight for irregular graphs. It develops the Decomposition Theorem, which imposes a weight-regular structure on Hoffman colorings with at least three colors, and the Composition Theorem, enabling construction of larger Hoffman colorable graphs from templates. These tools yield complete classifications for the Hoffman colorability of cone graphs and line graphs, and underpin an algorithm to enumerate all connected Hoffman colorable graphs with a given and , with empirical results up to six colors. As a byproduct, the authors determine values of several coloring parameters lying between Hoffman's bound and the chromatic number, and provide extensive computational data illustrating the landscape of 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.
Paper Structure (16 sections, 21 theorems, 17 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 16 sections, 21 theorems, 17 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perron-Frobenius and positive eigenvectors
  4. Interlacing
  5. Equitable partitions and quotients
  6. Weight-equitable partitions and -quotients
  7. The Decomposition Theorem and its consequences
  8. The Decomposition Theorem
  9. Some preliminary corollaries of the Decomposition Theorem
  10. Hoffman colorability of cone graphs
  11. Hoffman colorability of line graphs
  12. The Composition Theorem
  13. The Composition Theorem
  14. Regular Hoffman colorable graphs
  15. The algorithm
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $G$ is regular with $n$ vertices and largest and smallest adjacency eigenvalues $\lambda_{\max},\lambda_{\min}$ respectively, then and if a coclique $C$ meets this bound, then every vertex not in $C$ is adjacent to precisely $-\lambda_{\min}$ vertices of $C$.

Figures (4)

  • Figure 1: Irregular graph meeting Hoffman's bound: $n=9$, $\alpha=5$, $\lambda_{\min}=-2, \lambda_{\max}=4$, $\chi=3$.
  • Figure 2: The cone graphs over the 8-cycle and the disjoint union of two 4-cycles.
  • Figure 3: Two maximal alternating paths in a tree.
  • Figure 5: A pair of color complemented graphs.

Theorems & Definitions (32)

  • Theorem 1.1: Ratio bound, unpublished; see spectra
  • Theorem 1.2: Hoffman's bound, Hoffman
  • Theorem 2.1: Interlacing Theorem, spectra
  • Corollary 2.2: Cauchy interlacing, spectra
  • Corollary 2.3: Quotient matrix interlacing spectra
  • Proposition 2.4: 3chromDRG
  • Corollary 2.5: Fiol
  • Proposition 2.6: Abiad
  • Theorem 3.1
  • Lemma 3.2
  • ...and 22 more