Abstract
Let $G$ be a graph and let $c: V(G)\to\binom{[5]}{2}$ be an assignment of $2$-element subsets of the set $[5]$ to the vertices of $G$ such that for every edge $vw$, the sets $c(v)$ and $c(w)$ are disjoint. We call such an assignment a {\em $(5,2)$-coloring}. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The {\em odd-girth} of a graph $G$ is the length of the shortest odd cycle in $G$ ($\infty$ if $G$ is bipartite). We prove that every planar graph of odd-girth at least $9$ is $(5,2)$-colorable, and thus it is homomorphic to the Petersen graph. Also, this implies that such graphs have fractional chromatic number at most $5\over2$. As a special case, this result holds for planar graphs of girth at least $8$.