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 π(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 π(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) = π(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 π(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. Keywords-: DAG (Directed acyclic graphs); load; wavelengths; dipaths; good labelings; conflict graphs; intersection graphs; chromatic number.