Computation of lucky number of planar graphs is NP-hard

Abstract

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, ∑ w ∼ v ℓ ( w ) ≠ ∑ w ∼ u ℓ ( w ) ( x ∼ y means that x is joined to y). A lucky number of G, denoted by η ( 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 η ( G ) = 2 is NP-complete. Also for every k ⩾ 2 , it is NP-complete to decide whether η ( G ) = k for a given graph G.