Circular Coloring and Fractional Coloring in Planar Graphs

TL;DR

This paper investigates the conditions under which planar graphs with certain girth constraints admit circular and fractional colorings, establishing new bounds and conjectures related to Steinberg's and Jaeger's conjectures.

Contribution

It proves that the function f(k) exists only for odd prime k, provides bounds for f(p), and supports a related fractional coloring conjecture for planar graphs.

Findings

f(k) exists iff k is an odd prime

Bounds for f(p) are established for primes p≥5

A fractional coloring result for planar graphs with girth constraints

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 $k\ge 5$.