A sigma partitioning of a graph G is a partition of the vertices
into sets P1​,…,Pk​ such that for every two adjacent vertices u and
v there is an index i such that u and v have different numbers of
neighbors in Pi​. The  sigma number of a graph G, denoted by
σ(G), is the minimum number k such that G has a sigma partitioning
P1​,…,Pk​. Also, 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)î€ =∑w∼u​ℓ(w) (x∼y means that x and y are adjacent). The  lucky number of G, denoted by η(G), is the minimum number k such
that G has a lucky labeling ℓ:V(G)→Nk​. It was
conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is NP-complete to decide whether η(G)=2 for a given 3-regular
graph G. In this work, we prove this conjecture. Among other results, we give
an upper bound of five for the sigma number of a uniformly random graph