Injective colorings of sparse graphs

Let mad ( G ) denote the maximum average degree (over all subgraphs) of G and let χ i ( G ) denote the injective chromatic number of G . We prove that if mad ( G ) ≤ 5 2 , then χ i ( G ) ≤ Δ ( G ) + 1 ; and if mad ( G ) < 42 19 , then χ i ( G ) = Δ ( G ) . Suppose that G is a planar graph with girth g ( G ) and Δ ( G ) ≥ 4 . We prove that if g ( G ) ≥ 9 , then χ i ( G ) ≤ Δ ( G ) + 1 ; similarly, if g ( G ) ≥ 13 , then χ i ( G ) = Δ ( G ) .