Colouring t-perfect graphs (2412.17735v1)

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, Chv\'{a}tal 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 $\omega$ is $(\omega + 199050)$-colourable.

• The paper establishes the first finite chromatic bound on t-perfect graphs by proving they are 199,053-colourable.
• It extends the results to h-perfect graphs by showing every graph with clique number ω is (ω + 199,050)-colourable.
• The study introduces advanced tools like t-minor operations and arithmetic ropes, offering new structural insights in graph theory.

An Analysis of Colouring t-Perfect Graphs

The paper "Colouring t-perfect graphs" explores several aspects of the chromatic and structural properties of t-perfect graphs, expanding upon important results within the domain of graph theory. The authors provide novel insights into the colouring of these graphs while addressing long-standing open problems and conjectures in the field.

Key Contributions

1. Chromatic Number of t-Perfect Graphs: The paper establishes the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199,053-colourable. This significant bound addresses a question by Shepherd from 1995 and represents a major step in understanding the colourability properties of t-perfect graphs.
2. Bound for h-Perfect Graphs: The research extends beyond t-perfect graphs to h-perfect graphs, demonstrating that any h-perfect graph with a clique number $\omega$ is $(\omega + 199,050)$-colourable. This result provides a $\chi$-bound for h-perfect graphs, classifying them as $\chi$-bounded relative to their clique number.
3. Structural Insights and Techniques: The paper employs several advanced techniques, including t-minor operations and arithmetic ropes, to deduce the chromatic properties of the graphs under consideration. The introduction of the arithmetic rope as a structural tool is of particular interest, offering a novel approach to identifying critical substructures within large chromatic number graphs.
4. Connection to Perfect Graphs: The research effectively bridges the properties of perfect and t-perfect graphs, leveraging key theorems about perfect graphs to extrapolate properties of t-perfect graphs. This connection is foundational to understanding how t-perfect graphs expand the concept of perfect graphs.
5. Extension to $\overline{h}$-Perfect Graphs: The paper conjectures that the class of graphs that are complements of h-perfect graphs is also $\chi$-bounded and provides a quadratic bound for these complements, suggesting potential for further refinement.

Implications and Future Directions

The findings have profound implications for both theoretical graph theory and practical applications that hinge on understanding graph colourability. Firstly, by constraining the chromatic number of t-perfect graphs, this research provides foundational knowledge aiding algorithmic graph theory, particularly in the development of efficient algorithms for graph colouring problems.

Furthermore, the methods introduced, especially concerning t-minors and arithmetic ropes, open pathways for future research focusing on refining and extending these structural tools. Potential developments could involve defining broader classes of t-minors or finding more efficient constructions of arithmetic ropes that minimize the gap between bounds and actual chromatic numbers.

Additionally, the paper posits intriguing open questions, notably around the existence of other 4-critical t-perfect graphs beyond the known examples. These questions extend the groundwork for investigating critical elements within other lesser-explored subfamilies of graphs.

Overall, this research advances critical knowledge in graph theory by addressing longstanding conjectures, introducing novel mathematical tools, and solidifying important connections between different graph classes. These contributions underline the rich potential for continued exploration and develop a greater understanding of graph theoretical phenomena in complex networks.