Group sum chromatic number of graphs

We investigate the group sum chromatic number   (χgΣ(G)) of graphs, i.e. the smallest value ss such that taking any Abelian group GG of order ss, there exists a function f:E(G)→Gf:E(G)→G such that the sums of edge labels properly colour the vertices. It is known that χgΣ(G)∈{χ(G),χ(G)+1} for any graph GG with no component of order less than 33 and we characterize the graphs for which χgΣ(G)=χ(G).