  We study the following Steinberg-type problem on circular coloring: for an
odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar
graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a
homomorphism to the odd cycle $C_k$ (or equivalently, is circular
$(k,\frac{k-1}{2})$-colorable). Known results and counterexamples on
Steinberg's Conjecture indicate that $f(3)\in\{6,7\}$. In this paper, we show
that $f(k)$ exists if and only if $k$ is an odd prime. Moreover, we prove that
for any prime $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$
We conjecture that $f(p)\le p^2-2p$, and observe that the truth of this
conjecture implies Jaeger's conjecture that every planar graph of girth $2p-2$
has a homomorphism to $C_p$ for any prime $p\ge 5$. Supporting this conjecture,
we prove a related fractional coloring result that every planar graph of girth
$k$ without cycles of length from $k+1$ to $\lfloor\frac{22k}{3}\rfloor$ is
fractional $(k:\frac{k-1}{2})$-colorable for any odd integer $k\ge 5$.
