On neighbour-distinguishing colourings from lists

Let G be a finite simple graph, let C be a set of colours (in this paper we shall always suppose C ⊆ N) and let f : E(G) → C be an edge colouring of G. The colour set of a vertex v ∈ V (G) with respect to f is the set Sf (v) of colours of edges incident to v. The colouring f is neighbour-distinguishing if it distinguishes any two adjacent vertices by their colour sets, i.e., Sf (u) 6= Sf (v) whenever u, v ∈ V (G) and uv ∈ E(G). As usual, we are interested in the smallest number of colours in a neighbour-distinguishing colouring of G. If the optimisation runs over all proper colourings, that number is called the neighbour-distinguishing index of G, in symbols ndi(G). In the general case (when colourings are not required to be proper) the corresponding invariant is known as the general neighbour-distinguishing index of G and denoted by gndi(G). The main goal of this paper is to analyse the list versions of the above problems. Denote by Pk(X) the set of all k-subsets of a set X and by Lk(G) the set of all mappings L : E(G) → Pk(N). The set L(e), L ∈ Lk(G), e ∈ E(G), is the list of available colours for the edge e. Given L ∈ Lk(G), the graphG is Lneighbour-distinguishable provided there is a proper neighbour-distinguishing colouring f : E(G) → N satisfying f(e) ∈ L(e) for every e ∈ E(G). The list neighbour-distinguishing index of G, in symbols lndi(G), is the minimum k such that for every L ∈ Lk(G) the graph G is L-neighbour-distinguishable. †Email: [email protected]. The work was supported by Science and Technology Assistance Agency under the contract No. APVV-0023-10, by Grant VEGA 1/0652/12 and by the Agency of the Slovak Ministry of Education for the Structural Funds of the EU under the project ITMS:26220120007. ‡Email: [email protected]. The research was partially supported by the Polish Ministry of Science and Higher Education.