  We study the problem of finding homomorphisms into odd cycles from planar
graphs with high odd-girth. The Jaeger-Zhang conjecture states that every
planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd
cycle $C_{2k+1}$. The $k=1$ case is the well-known Gr\"otzsch's $3$-coloring
theorem. For general $k$, in 2013 Lov\'asz, Thomassen, Wu, and Zhang showed
that it suffices to have odd-girth at least $6k+1$. Improvements are known for
$C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For
$C_9$ we improve this hypothesis by showing that it suffices to have odd-girth
23. Our main tool is a variation on the potential method applied to modular
orientations. This allows more flexibility when seeking reducible
configurations. The same techniques also prove some results on circular
coloring of signed planar graphs.
