NPC PROBLEM minimum sum of vertex coloring


For a graph G and a legal vertex-colouring ψ : V(G)→N of G,

let σψ(G) be the sum of ψ(v),

and set σ(G):=min σψ(G),

where the minimum ranges over all valid vertex colourings of G.

Prove that {(G,k): σ(G)≤ k}∈ NPC.

I thought of reduction from subset-sum but my problem is with the min and not with the sum, I don’t understand how to do a reduction to check the minimum. I got a hint to do a reduction from 3-NAE-SAT but I don’t understand how. would appreciate some help. even some source I can read about this problem if you know about something.