Monochromatic absorbency and independence in 3-quasi-transitive digraphs

Abstract

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. In an m-coloured digraph D we say that a subdigraph H is: monochromatic whenever all of its arcs are coloured alike, and almost monochromatic if with at most 1 exception all of its arcs are coloured with the same colour.If D is an m-coloured digraph a kmp or a kernel by monochromatic paths of D is a set K of vertices of D which is independent by monochromatic paths (for any two different vertices u, v ∊ K there are no monochromatic paths between them) such that for every other vertex x ∊ V(D) \\ K there is a vertex v ∊ K such that there is an xv monochromatic directed path in D.A digraph D is 3-quasi-transitive if whenever (x, y), (y, w), and (w, z) ∊ A(D) with x, y, w and z pairwise different vertices, either (x, z) or (z, x) is in A(D), and it is asymmetric if it has no symmetric arcs.In 1982, Sands, Sauer, and Woodrow proved that every 2-coloured tournament has a kmp. They also posed the following problem: Let T be a 3-coloured tournament which does not contain Ĉ3 (the 3-coloured cyclic tournament of order 3). Then, must T contain a kmp?In this paper we consider asymmetric 3-quasi-transitive digraphs, which not only generalise tournaments but also bipartite tournaments, and prove that if D is an m-coloured asymmetric 3-quasi-transitive digraph such that every C4 (the directed cycle of length 4) is monochromatic and every C3 (the directed cycle of length 3) is almost monochromatic, then D has a kernel by monochromatic paths.We also note that the hypotheses on C3 and C4 are tight.