Improved bounds for colouring circle graphs

We prove the first χ χ -bounding function for circle graphs that is optimal up to a constant factor. To be more precise, we prove that every circle graph with clique number at most ω ω has chromatic number at most 2 ω log 2 ⁡ ( ω ) + 2 ω log 2 ⁡ ( log 2 ⁡ ( ω ) ) + 10 ω 2ω log _2 (ω ) +2ω log _2(log _2 (ω )) + 10ω .