Computation of lucky number of planar graphs is NP-hard

A lucky labeling of a graph G is a function l : V(G) --> N, such that for every two adjacent vertices v and u of G, Sigma(w similar to v) l(w) not equal Sigma(w similar to u) l(w) (x similar to y means that x is joined to y). A lucky number of G, denoted by eta(G), is the minimum number k such that G has a lucky labeling l : V(G) --> {1, ..., k}. We prove that for a given planar 3-colorable graph G determining whether eta(G) = 2 is NP-complete. Also for every k >= 2, it is NP-complete to decide whether eta(G) = k for a given graph G. (C) 2011 Elsevier B.V. All rights reserved.