Circular coloring and fractional coloring in planar graphs

Abstract

AbstractWe study the following Steinberg‐type problem on circular coloring: for an odd integer , what is the smallest number such that every planar graph of girth without cycles of length from to admits a homomorphism to the odd cycle (or equivalently, is circular ‐colorable). Known results and counterexamples on Steinberg's Conjecture indicate that . In this paper, we show that exists if and only if is an odd prime. Moreover, we prove that for any prime , We conjecture that , and observe that the truth of this conjecture implies Jaeger's conjecture that every planar graph of girth has a homomorphism to for any prime . Supporting this conjecture, we prove a related fractional coloring result that every planar graph of girth without cycles of length from to is fractional ‐colorable for any odd integer .