Properly orderable graphs Original Research Article

In a graph G = (V, E) provided with a linear order ‘ < ’ on V, a chordless path with vertices a, b, c, d and edges ab, bc, cd is called an obstruction if both a < b and d < c hold. Chvátal (1984) defined the class of perfectly orderable graphs (i.e., graphs possessing an acyclic orientation of the edges such that no obstruction is induced) and proved that they are perfect. We introduce here the class of properly orderable graphs which is a generalization of Chvátalʹs class of perfectly orderable graphs: obstructions are forbidden only in the subgraphs induced by the vertices of an odd cycle. We prove the perfection of these graphs and give an O(m2 + mn + n) colouring algorithm.