Homomorphisms from sparse graphs to the Petersen graph

Let G be a graph and let c : V ( G ) → ( { 1 , … , 5 } 2 ) be an assignment of 2 -elements subsets of the set { 1 , … , 5 } to the vertices of G such that for any two adjacent vertices u and v , c ( u ) and c ( v ) are disjoint. Call such a coloring c a (5, 2)-coloring of G . A graph is ( 5 , 2 ) -colorable if and only if it has a homomorphism to the Petersen graph. ximum average degree of G 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 ) < 5 2 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.