Cayley graphs of abelian groups, circular colorings and discrete fundamental groups

For each integer $k\ge 2$, we prove that an abelian group $\mathrm{\Gamma }$ satisfies $\exp (\mathrm{\Gamma })\in \{1,\dots ,k\}\cup \{2,4,\dots ,2k\}$ if and only if it admits no non-bipartite Cayley graph whose circular chromatic number is strictly less than $2+\frac{2}{k-1}$. This refines the algebraic characterization by Krebs and Sankar of abelian groups that admit $3$-chromatic Cayley graphs.