Directed acyclic graphs with the unique dipath property

Let P be a family of dipaths of a DAG (Directed Acyclic Graph) G. The load of an arc is the number of dipaths containing this arc. Let @p(G,P) be the maximum of the load of all the arcs and let w(G,P) be the minimum number of wavelengths (colors) needed to color the family of dipaths P in such a way that two dipaths with the same wavelength are arc-disjoint. There exist DAGs such that the ratio between w(G,P) and @p(G,P) cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contains neither a source nor a sink of G). We prove that, for any family of dipaths P, w(G,P)=@p(G,P) if and only if G is without internal cycle. We also consider a new class of DAGs, called UPP-DAGs, for which there is at most one dipath from a vertex to another. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between w(G,P) and @p(G,P) cannot be bounded (a result conjectured in an other article). For that we introduce ''good labelings'' of the conflict graph associated to G and P, namely labelings of the edges such that for any ordered pair of vertices (x,y) there do not exist two paths from x to y with increasing labels.