用更少颜色为 t-完美图着色
Coloring t-perfect graphs with fewer colors
Matija Novaković, Stefan Weltge
AI 速览：通过改进既有证明，将 t-完美图的着色数上界从 199053 降低到 130。
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 130 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 to 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.