1 research outputs found
On 4-critical t-perfect graphs
It is an open question whether the chromatic number of -perfect graphs is
bounded by a constant. The largest known value for this parameter is 4, and the
only example of a 4-critical -perfect graph, due to Laurent and Seymour, is
the complement of the line graph of the prism (a graph is 4-critical if
it has chromatic number 4 and all its proper induced subgraphs are
3-colorable).
In this paper, we show a new example of a 4-critical -perfect graph: the
complement of the line graph of the 5-wheel . Furthermore, we prove that
these two examples are in fact the only 4-critical -perfect graphs in the
class of complements of line graphs. As a byproduct, an analogous and more
general result is obtained for -perfect graphs in this class.
The class of -free graphs is a proper superclass of complements of line
graphs and appears as a natural candidate to further investigate the chromatic
number of -perfect graphs. We observe that a result of Randerath,
Schiermeyer and Tewes implies that every -perfect -free graph is
4-colorable. Finally, we use results of Chudnovsky et al to show that
and are also the only 4-critical
-perfect -free graphs