χ-Diperfect digraphs

Abstract

Let D be a digraph. A coloring C and a path P of D are orthogonal if P contains exactly one vertex of each color class in C. In 1982, Berge defined the class of χ-diperfect digraphs. A digraph D is χ-diperfect if for every minimum coloring C of D, there exists a path P orthogonal to C and this property holds for every induced subdigraph of D. Berge showed that every symmetric digraph is χ-diperfect, as well as every digraph whose underlying graph is perfect. However, he also showed that not every super-orientation of an odd cycle or complement of an odd cycle is χ-diperfect. Non-χ-diperfect super-orientations of odd cycles and their complements may play an important role in the characterization of χ-diperfect digraphs, similarly to the role they play in the characterization of perfect graphs. In this paper, we present a characterization of super-orientations of odd cycles and a characterization of super-orientations of complements of odd cycles that are χ-diperfect. Moreover, we show that certain classes of digraphs that are free of such non-χ-diperfect structures are χ-diperfect.