Incidence coloring of graphs with high maximum average degree

An incidence of an undirected graph G is a pair (v, e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, (ii) e = f or (iii) vw = e or f . An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama et al. [6] proved that every graph with maximum average degree strictly less than 3 can be incidence colored with ∆+ 3 colors. Recently, Bonamy et al. [2] proved that every graph with maximum degree at least 4 and with maximum average degree strictly less than 7 3 admits an incidence (∆+1)-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between 4 and 6. In particular we prove that every graph with maximum degree at least 7 and with maximum average degree less than 4 admits an incidence (∆+3)-coloring. This result implies that every triangle-free planar graph with maximum degree at least 7 is incidence (∆ + 3)-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence (∆ + 7)-coloring. More generally, we prove that ∆+ k − 1 colors are enough when the maximum average degree is less than k and the maximum degree is sufficiently large.