On circle graphs with girth at least five

Circle graphs with girth at least five are known to be 2-degenerate [A.A. Ageev, Every circle graph with girth at least 5 is 3-colourable, Discrete Math. 195 (1999) 229–233]. In this paper, we prove that circle graphs with girth at least g≥5g≥5 and minimum degree at least two contain a chain of g−4g−4 vertices of degree two, which implies Ageev’s result in the case g=5g=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≥5g≥5 as well as a precise estimate of their maximum average degree.