Coloring circle graphs

Circle graph is an intersection graph of chords of a circle. We denote the class of circle graphs by cir. In this paper we investigate the chromatic number of the circle graph as a function of the size of maximum clique ω = ω ( G ) . More precisely we investigate f ( k ) = max { χ ( G ) | G ∈ CIR  & ω ( G ) ⩽ k } . Kratochvíl and Kostochka showed that f ( k ) ⩽ 50 ⋅ 2 k − 32 k − 64 . The best lower bound is by Kostochka and is f ( k ) = Ω ( k log k ) . We improve the upper bound to f ( k ) ⩽ 21 ⋅ 2 k − 24 k − 24 . We also present the bound χ ( G ) ⩽ ω ⋅ log n which shows, that the circle graphs with small maximum clique and large chromatic number must have many vertices.