On graphs all of whose {C3, T3}-free arc colorations are kernel-perfect

A digraph D is called a kernel-perfect digraph or KP -digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V (ζ(D)) = V (D), and A (ζ(D)) = ∪ i {(u, v) with colour i: there exists a monochromatic path of colour i from the vertex u to the vertex v contained in D}. We will denoted by T3 and C3, the transitive tournament of order 3 and the 3-directed-cycle respectively; both of whose arcs are coloured with three different colours. Let G be a simple graph. By an m-orientation-coloration of G we mean an m-coloured digraph which is an asymmetric orientation of G. By the class E we mean the set of all the simple graphs G that for any m-orientation-coloration D without C3 or T3, we have that ζ(D) is a KP -digraph. In this paper we prove that if G is a hamiltonian graph of class E, then its complement has at most one nontrivial component, and this component is K3 or a star.