Kernels in digraphs with covering number at most 3

AbstractA kernel N of a digraph D is an independent set of vertices of D such that for every x∈V(D)−N there exists an arc from x to N. A digraph D is M-oriented if each directed triangle has at least two symmetrical arcs. The covering number of a digraph D denoted by θ(D) is the minimum number of complete subdigraphs of D that partition V(D). Let D be an M-oriented digraph with θ(D)⩽3. In this paper the following result is proved: If each directed cycle C of length 5 satisfies at least one of the two following properties: (a) C has two diagonals (b) C has three symmetrical arcs. Then D has a kernel. This result generalizes the previous results and as a consequence the following interesting conjecture is proved (now known to be false in general) in case that D is a digraph with θ(D)⩽3.[Meyniel, 1976]Conjecture 1 [M. Blidia, P. Duchet, F. Maffray, On the orientation of Meyniel graphs, J. Graph Theory 18(7) (1994) 705–711; P. Duchet, Graphes Noyau-Parfaits, Ann. Discrete Math. 9 (1980) 93–101, North-Holland Publishing Company]. If every directed cycle of odd length contained in D possesses two diagonals, then D has a kernel.