Fractional Coloring of Planar Graphs of Girth Five

Abstract

A graph $G$ is $(a:b)$-colorable if there exists an assignment of $b$-element subsets of $\{1,\ldots,a\}$ to vertices of $G$ such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph $G$ and a vertex $x\in V(G)$, the graph $G$ has a set coloring $\varphi$ by subsets of $\{1,\ldots,6\}$ such that $|\varphi(v)|\geq 2$ for $v\in V(G)$ and $|\varphi(x)|=3$. As a corollary, every triangle-free planar graph on $n$ vertices is $(6n:2n+1)$-colorable. We further use this result to prove that for every $\Delta$, there exists a constant $M_{\Delta}$ such that every planar graph $G$ of girth at least five and maximum degree $\Delta$ is $(6M_{\Delta}:2M_{\Delta}+1)$-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree $\Delta$ have fractional chromatic number at most $3-\frac{3}{2M_{\Delta}+1}$.