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 gi...

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