On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate (Ageev, 1999). In this paper, we prove that circle graphs with girth at least g ⩾ 5 contain a vertex of degree at most one, or a chain of g− 4 vertices of degree two, which implies Ageev's result in the case g = 5. We then use this structural property to give an upper bound on the circular chromatic number of circle graphs with girth at least g ⩾ 5 as well as a precise estimate of their maximum average degree.