A class of β-perfect graphs

Consider the following total order: order the vertices by repeatedly removing a vertex of minimum degree in the subgraph of vertices not yet chosen and placing it after all the remaining vertices but before all the vertices already removed. For which graphs the greedy algorithm on this order gives an optimum vertex-coloring? Markossian, Gasparian and Reed introduced the class of β-perfect graphs. These graphs admit such a greedy vertex-coloring algorithm. The recognition of β-perfect graphs is open. We define a subclass of β-perfect graphs, that can be recognized in polynomial time, by considering the class of graphs with no even hole, no short-chorded cycle on six vertices, and no diamond. In particular, we make use of the following properties: no minimal β-imperfect graph contains a simplicial vertex, a minimal β-imperfect graph which is not an even hole contains no vertex of degree 2.