On the maximum average degree and the incidence chromatic number of a graph

We prove that the incidence chromatic number of every 3-degenerated graph G is at most Δ(G)+4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G)<3 is at most Δ (G)+3. We show that when Δ (G) ≥ 5, this bound may be decreased to Δ (G)+2. Moreover, we show that for every graph G with mad(G)<22/9 (resp. with mad(G)<16/7 and Δ (G)≥ 4), this bound may be decreased to Δ(G)+2 (resp. to Δ(G)+1).