Planar Graphs of Odd-Girth at Least 9 are Homomorphic to the Petersen Graph
Zdenĕk Dvořák, Riste Škrekovski, Tomáš Valla · SIAM Journal on Discrete Mathematics · 2008
Let G be a graph and let $c: V(G)\to\binom{1,\ldots,5}{2}$ be an assignment of 2-element subsets of the set $1,\ldots,5$ 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 ($\infty$ 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 a fractional chromatic number at most $5\over2$. As a special case, this result holds for planar graphs of girth at least 8.