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Every Circle Graph of Girth at Least 5 is 3Colourable A. A. Ageev \Lambda

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
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 Ĝ(\Gamma k ) = fmaxĜ(G) : G 2
\Gamma k ) where Ĝ(G) stands for the chromatic number of G. Since \Gamma 3
contains all complete
graphs, Ĝ(\Gamma 3 ) = 1. The problem of evaluating Ĝ(\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 Ĝ(\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


Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk


Collections: Mathematics