**Problem**Does every triangle-free planar graph of maximum degree three have circular chromatic number at most ?

Throughout, we let denote the circular chromatic number of the graph .

A well-known Question of Nesetril asks if for all cubic graphs of sufficiently high girth. A conjecture of Jaeger asserts that for every planar graph of girth . There are numerous partial results on these problems, and there are many interesting questions concerning the circular chromatic numbers of restricted families of graphs. Here we are restricted to planar graphs of girth with maximum degree . The dodecahedron lives in this class and has . It remains unclear if anyone else in this class might have larger.

A related conjecture of X. Zhu asserts that for every triangle-free planar graph with and one has .

## Bibliography

* indicates original appearance(s) of problem.