Journal of Graph Theory | 2021
Circular coloring and fractional coloring in planar graphs
Abstract
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 $k\\ge 5$.