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It was conjectured by Jaeger that $4k$-edge connected graphs admit a $(2k+1, k)$-flow. The restriction of this conjecture to planar graphs is equivalent to the statement that planar graphs of girth at least $4k$ have circular chromatic number at most $2+ {{1}\over {k}}$. Even this restricted version of Jaeger's conjecture is largely open. The $k=1$ case is the well-known Grotzsch 3-colour theorem. This paper proves that for $k \geq 2$, planar graphs of odd girth at least $8k-3$ have circular chromatic number at most $2+{{1}\over {k}}$.

Circular Chromatic Number of Planar Graphs of Large Odd Girth