Group Sum Chromatic Number of Graphs

We investigate the $\mathit{\text{group}}\mathit{\text{sum}}\mathit{\text{chromatic}}\mathit{\text{number}}$ ($\text{gchi}(G)$) of graphs, i.e. the smallest value $s$ such that taking any Abelian group $\text{gr}$ of order $s$, there exists a function $f:E(G)\to \text{gr}$ such that the sums of edge labels properly colour the vertices. It is known that $\text{gchi}(G)\in \{\chi (G),\chi (G)+1\}$ for any graph $G$ with no component of order less than $3$ and we characterize the graphs for which $\text{gchi}(G)=\chi (G)$.