Injective colorings of sparse graphs

Let mad(G) denote the maximum average degree (over all subgraphs) of GG and let χi(G)χi(G) denote the injective chromatic number of GG. We prove that if mad(G)≤52, then χi(G)≤Δ(G)+1χi(G)≤Δ(G)+1; and if mad(G)<4219, then χi(G)=Δ(G)χi(G)=Δ(G). Suppose that GG is a planar graph with girth g(G)g(G) and Δ(G)≥4Δ(G)≥4. We prove that if g(G)≥9g(G)≥9, then χi(G)≤Δ(G)+1χi(G)≤Δ(G)+1; similarly, if g(G)≥13g(G)≥13, then χi(G)=Δ(G)χi(G)=Δ(G).