Homomorphisms from sparse graphs to the Petersen graph

Let GG be a graph and let cc: V(G)→({1,…,5}2)be an assignment of 22-elements subsets of the set {1,…,5}{1,…,5} to the vertices of GG such that for any two adjacent vertices uu and v,c(u)v,c(u) and c(v)c(v) are disjoint. Call such a coloring cc a (5, 2)-coloring of GG. A graph is (5,2)(5,2)-colorable if and only if it has a homomorphism to the Petersen graph.The maximum average degree   of GG is defined as Mad(G)=max{2|E(H)||V(H)|:H⊆G}. In this paper, we prove that every triangle-free graph with Mad(G)<52 is homomorphic to the Petersen graph. In other words, such a graph is (5, 2)-colorable. Moreover, we show that the bound on the maximum average degree in our result is best possible.