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Spectral approaches for $d$-improper chromatic number

Krystal Guo, Ross J. Kang, Gabriëlle Zwaneveld

Abstract

In this paper, we explore algebraic approaches to $d$-improper and $t$-clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a generalization of the Hoffman bound for $d$-improper colourings. We strengthen this theorem by characterizing the equality case. In particular, if the Hoffman bound is tight for a graph $G$, then the $d$-improper Hoffman bound is tight for the strong product $G \boxtimes K_{d+1}$. Moreover, we prove d-improper analogous for the inertia bound by Cvetkovíc and the multi-eigenvalue lower bounds of Elphick and Wocjan. We conjecture an equality between the chromatic number of a graph $G$ and the $d$-improper chromatic number of its strong product with a complete graph, $G \boxtimes K_{d+1}$, and prove the conjecture in special graph classes, including perfect graphs and graphs with chromatic number at most 4. Other supporting evidence for the conjecture includes a fractional analogue, a clustered analogue, and various spectral relaxations of the equality.

Spectral approaches for $d$-improper chromatic number

Abstract

In this paper, we explore algebraic approaches to -improper and -clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a generalization of the Hoffman bound for -improper colourings. We strengthen this theorem by characterizing the equality case. In particular, if the Hoffman bound is tight for a graph , then the -improper Hoffman bound is tight for the strong product . Moreover, we prove d-improper analogous for the inertia bound by Cvetkovíc and the multi-eigenvalue lower bounds of Elphick and Wocjan. We conjecture an equality between the chromatic number of a graph and the -improper chromatic number of its strong product with a complete graph, , and prove the conjecture in special graph classes, including perfect graphs and graphs with chromatic number at most 4. Other supporting evidence for the conjecture includes a fractional analogue, a clustered analogue, and various spectral relaxations of the equality.

Paper Structure

This paper contains 9 sections, 26 theorems, 72 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Preliminary observations
  4. Spectral bounds for the d-improper chromatic number
  5. d-improper Hoffman colourings
  6. Other spectral bounds
  7. The d-improper chromatic number of G box complete d+1
  8. The c-clustered chromatic numbers of G box complete graph
  9. Generalizing bounds of Wocjan and Elphick

Key Result

Theorem 1

For every integer $d\geq 0$ and every graph $G$, we have

Figures (6)

  • Figure 1: Colourings of $G$ can be lifted to $d$-improper colourings of $G \boxtimes K_t$, but not all $d$-improper colourings of $G \boxtimes K_{d+1}$ are lifted from some colouring of $G$. In the black and white version, the colours are represented by different shades of gray.
  • Figure 2: Different colourings of the Petersen graph. In the black and white version, the colours are represented by different shades of gray.
  • Figure 3: A $2$-improper colouring with 5 colours of $K_6[{\mkern2mu\overline{\mkern-2muK_3}}]$ drawn in its complement ${\mkern2mu\overline{\mkern-2muK_6[{\mkern2mu\overline{\mkern-2muK_3}}]}} = 6 K_3$. In the colouring of the complement graph, every vertex is non-adjacent to (at most) two vertices of its colour.
  • Figure 4: A $3$-clustered edge colouring of Paley(9) with 2 colours.
  • Figure 5: The Bowtie graph with a $1$-improper independent set.
  • ...and 1 more figures

Theorems & Definitions (62)

  • Theorem 1: Bilu BILU2006608
  • Conjecture 1
  • Remark
  • Lemma 2: Lem. 14 in EsWo23+
  • proof
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • ...and 52 more