Planar Graphs of Odd-Girth at Least 9 are Homomorphic to the Petersen Graph

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 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 (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd cycle in G (∞ 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 2 . As a special case, this result holds for planar graphs of girth at least 8.