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)≤Δ(G)+1; and if mad(G)<42/19, then χi​(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
χi​(G)≤Δ(G)+1; similarly, if g(G)≥13, then
χi​(G)=Δ(G).Comment: 10 page