11 research outputs found
Perfectly contractile graphs and quadratic toric rings
Perfect graphs form one of the distinguished classes of finite simple graphs.
In 2006, Chudnovsky, Robertson, Saymour and Thomas proved that a graph is
perfect if and only if it has no odd holes and no odd antiholes as induced
subgraphs, which was conjectured by Berge. We consider the class
of graphs that have no odd holes, no antiholes and no odd stretchers as induced
subgraphs. In particular, every graph belonging to is perfect.
Everett and Reed conjectured that a graph belongs to if and only
if it is perfectly contractile. In the present paper, we discuss graphs
belonging to from a viewpoint of commutative algebra. In fact,
we conjecture that a perfect graph belongs to if and only if
the toric ideal of the stable set polytope of is generated by quadratic
binomials. Especially, we show that this conjecture is true for Meyniel graphs,
perfectly orderable graphs, and clique separable graphs, which are perfectly
contractile graphs.Comment: 10 page