Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Colouring t-perfect graphs

Maria Chudnovsky, Linda Cook, James Davies, Sang-il Oum, Jane Tan

TL;DR

This work studies the finite colourability of $t$-perfect graphs, the class characterized by $\mathrm{SSP}(G)=\mathrm{TSTAB}(G)$ with stabilizing odd-cycle inequalities. The authors prove that every $t$-perfect graph is $199053$-colourable, and as a corollary any $h$-perfect graph with clique number $\omega$ is $(\omega+199050)$-colourable, establishing a linear-$\chi$ bound for these classes. The proof combines a reduction to graphs with odd girth at least $11$, a fragmentation into levels to locate high-chromatic induced subgraphs, and a novel use of $5$-arithmetic ropes to force an odd wheel $t$-minor via $t$-contractions, which is forbidden in $t$-perfect graphs. These results place $t$- and $h$-perfect graphs within the realm of $\chi$-bounded classes and resolve Shepherd’s long-standing question in the finite-bound sense, with implications for polyhedral descriptions and algorithmic colourability.

Abstract

Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chvátal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are $199053$-colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995. Our proof also shows that every h-perfect graph with clique number $ω$ is $(ω+ 199050)$-colourable.

Colouring t-perfect graphs

TL;DR

This work studies the finite colourability of -perfect graphs, the class characterized by with stabilizing odd-cycle inequalities. The authors prove that every -perfect graph is -colourable, and as a corollary any -perfect graph with clique number is -colourable, establishing a linear- bound for these classes. The proof combines a reduction to graphs with odd girth at least , a fragmentation into levels to locate high-chromatic induced subgraphs, and a novel use of -arithmetic ropes to force an odd wheel -minor via -contractions, which is forbidden in -perfect graphs. These results place - and -perfect graphs within the realm of -bounded classes and resolve Shepherd’s long-standing question in the finite-bound sense, with implications for polyhedral descriptions and algorithmic colourability.

Abstract

Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chvátal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are -colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995. Our proof also shows that every h-perfect graph with clique number is -colourable.

Paper Structure

This paper contains 13 sections, 21 theorems, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Connection with $\chi$-boundedness
  4. Our approach
  5. Extracting an odd wheel t-minor
  6. Organisation
  7. Preliminaries
  8. Reducing to the case where there are no short odd cycles
  9. Remarks on \ref{['thm:technical-main']}
  10. Arithmetic ropes
  11. Odd wheel t-minors
  12. Concluding remarks and open problems
  13. Acknowledgements.

Key Result

Theorem 1.1

Every t-perfect graph is $199053$-colourable.

Figures (4)

  • Figure 1: The only known $4$-critical t-perfect graphs in the literature. On the left is the complement of the line graph of the complement of $C_6$ found by Laurent and Seymour schrijver2003combinatorial, and on the right is the complement of the line graph of the $5$-wheel found by Benchetrit benchetrit-PhDbenchetrit20164critical.
  • Figure 2: A $5$-arithmetic rope. Dotted lines represent odd-length paths and dashed lines represent even-length paths.
  • Figure 3: The situation in the proof of \ref{['lem:ropeinduction']} when $\chi(C_0)\geqslant c+3$.
  • Figure 4: The situation in the proof of \ref{['lem:ropeinduction']} when $\chi(C_0)<c+3$ and $\chi(C_1)\geqslant c+3$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Gerards and Shepherd GS1998
  • proof
  • Lemma 2.2: Folklore; see benchetrit-PhD
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 31 more