Generalized list colourings of graphs

1. Introduction and NotationWe consider finite undirected graphs without loops and multiple edges. Thevertex set of a graph G is denoted by V(G) and the edge set by E(G). Thenotation H ⊆ G means that H is a subgraph of G. The vertex induced (wewill say briefly: induced) subgraph H of G is denoted by H ≤G. We say thatG contains H whenever G contains a subgraph isomorphic to H. In general,we follow the notation and terminology of [5].Let I denote the set of all mutually nonisomorphic graphs. If P is anonempty subset of I, then P will also denote the property that a graph is