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Linear colorings of graphs

Claire Hilaire, Matjaž Krnc, Martin Milanič, Jean-Florent Raymond

TL;DR

This work analyzes the linear chromatic number $\chi_{\mathrm{lin}}$ as a relaxation of treedepth, comparing it to the centered chromatic number $\chi_{\mathrm{cen}}$ and exploring its algorithmic implications. It establishes sharp bounds across key graph classes, proving $\chi_{\mathrm{cen}}(G) \leq c\cdot \chi_{\mathrm{lin}}(G)^2$ for all proper minor-closed classes and obtaining linear bounds for graphs of bounded treewidth, among other results. The paper also shows that in several natural graph families, including caterpillars, complete multipartite graphs, and complements of rook's graphs, $\chi_{\mathrm{lin}}$ and $\chi_{\mathrm{cen}}$ coincide or differ by only a small additive constant, and it provides finite obstructions for graphs with bounded $\chi_{\mathrm{lin}}$, along with NP-hardness and fixed-parameter tractability results for computing $\chi_{\mathrm{lin}}$. These contributions advance understanding of how linear colorings approximate treedepth and inform algorithmic approaches for bounded-expansion and related graph classes. The work also highlights open questions, notably whether treedepth can be bounded by a constant multiple of $\chi_{\mathrm{lin}}$ in general and the precise behavior of these invariants on trees.

Abstract

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that treedepth could be bounded from above by twice the linear chromatic number. In this paper we investigate the properties of linear chromatic number and provide improved bounds in several graph classes.

Linear colorings of graphs

TL;DR

This work analyzes the linear chromatic number as a relaxation of treedepth, comparing it to the centered chromatic number and exploring its algorithmic implications. It establishes sharp bounds across key graph classes, proving for all proper minor-closed classes and obtaining linear bounds for graphs of bounded treewidth, among other results. The paper also shows that in several natural graph families, including caterpillars, complete multipartite graphs, and complements of rook's graphs, and coincide or differ by only a small additive constant, and it provides finite obstructions for graphs with bounded , along with NP-hardness and fixed-parameter tractability results for computing . These contributions advance understanding of how linear colorings approximate treedepth and inform algorithmic approaches for bounded-expansion and related graph classes. The work also highlights open questions, notably whether treedepth can be bounded by a constant multiple of in general and the precise behavior of these invariants on trees.

Abstract

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that treedepth could be bounded from above by twice the linear chromatic number. In this paper we investigate the properties of linear chromatic number and provide improved bounds in several graph classes.
Paper Structure (23 sections, 36 theorems, 7 equations, 4 figures, 1 table)

This paper contains 23 sections, 36 theorems, 7 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminaries
  4. Notations and definitions
  5. Basics.
  6. Graph containment relations.
  7. Tree decompositions.
  8. Colorings.
  9. Centered versus linear
  10. General bounds
  11. When centered is almost linear
  12. Graphs for which every linear coloring is centered
  13. Case 1.
  14. Case 2.
  15. Complete multipartite graphs and complements of rook's graphs
  16. ...and 8 more sections

Key Result

Theorem 1.1

Every graph $G$ satisfies $\mathop{\mathrm{\chi_\mathrm{cen}}}\nolimits(G)\leqslant \mathop{\mathrm{\chi_\mathrm{lin}}}\nolimits(G)^{10} \cdot (\log \mathop{\mathrm{\chi_\mathrm{lin}}}\nolimits(G))^{\mathop{\mathrm{\mathcal{O}}}\nolimits(1)}$.

Figures (4)

  • Figure 1: A graph $G$ with $\mathop{\mathrm{\chi_\mathrm{lin}}}\nolimits(G)=4$ (left) and a topological minor $H$ of $G$ (right) with $\mathop{\mathrm{\chi_\mathrm{lin}}}\nolimits(H)>4$.
  • Figure 2: The coloring of \ref{['rem:corook']}
  • Figure 3: The coloring of \ref{['rem:op']}.
  • Figure 4: The list $\mathcal{F}$ of subgraph obstructions to $\mathop{\mathrm{\chi_\mathrm{lin}}}\nolimits(G)\leqslant 3$.

Theorems & Definitions (82)

  • Theorem 1.1: bose2022linear
  • Conjecture 1.1: kun2021polynomial
  • Theorem 1.2: kun2021polynomial
  • Theorem 1.3: kun2021polynomial
  • Remark 2.1
  • Proposition 2.2
  • proof
  • proof
  • Proposition 2.6
  • proof
  • ...and 72 more