Coloring t-perfect graphs with fewer colors

2026-07-07

AI Summary

This study addresses the problem of excessively high upper bounds on the chromatic number of t-perfect graphs. By refining the existence proof for r-arithmetic webs, the authors significantly reduce the number of colors required for proper coloring. The core of their approach lies in lowering the chromatic threshold guaranteeing the existence of r-arithmetic webs from exponential to linear order and introducing a weakened notion of arithmetic webs. Their construction integrates odd-girth analysis, t-minor theory, and hierarchical graph structures. As a result, this work improves the best-known upper bound on the chromatic number of t-perfect graphs from 199,053 to 186, representing a substantial advancement in the coloring theory of t-perfect graphs.

Abstract

Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 186 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction of t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.

Problem

• t-perfect graphs
• chromatic number
• coloring
• odd girth
• graph coloring

Innovation

• t-perfect graphs
• chromatic number
• arithmetic rope
• odd girth
• t-minor