 
Summary: Every Circle Graph of Girth at Least 5 is 3Colourable
A. A. Ageev \Lambda
Institute of Mathematics
Siberian Branch of Russian Academy of Sciences
Abstract
It is known that every trianglefree (equivalently, of girth at least 4) circle graph is
5colourable (Kostochka, 1988) and that there exist examples of these graphs which are
not 4colourable (Ageev, 1996). In this note we show that every circle graph of girth at
least 5 is 2degenerate and, consequently, not only 3colourable but even 3choosable.
Keywords: circle graph, chromatic number, girth
1 Introduction
The girth of a graph G is the length of a shortest circuit of G. A graph G is a circle graph
if it is isomorhic to the intersection graph of chords of a circle. Let \Gamma k (k = 3; 4; : : : )
denote the family of circle graphs with girth at least k and let Ø(\Gamma k ) = fmaxØ(G) : G 2
\Gamma k ) where Ø(G) stands for the chromatic number of G. Since \Gamma 3
contains all complete
graphs, Ø(\Gamma 3 ) = 1. The problem of evaluating Ø(\Gamma 4 ) (the maximum of chromatic
number over all trianglefree circle graphs) has been independently posed in [2] and
[4] (see also [3], p. 158). The ultimate result is that Ø(\Gamma 4 ) = 5; the upper bound is
due to A. Kostochka [5], the lower bound to the author [1]. In this note we prove that
