Injective colorings of sparse graphs

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