A triangle-free circle graph with chromatic number 5

Abstract It follows from the results of Karapetyan (1984), Gyarfas and Lehel (1985), and Kostochka (1988) that 4⩽χ* ⩽5 where χ* = max{χ(G): G is a triangle-free circle graph. We show that χ* ⩾ 5 and thus χ* = 5. This disproves the conjecture of Karapetyan that 4 ≤ χ * ≤ 5 and answers negatively a question of Gyarfas and Lehel.