Incidence coloring of planar graphs without adjacent small cycles

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 2012, Yang [15] proved that every planar graph has an incidence coloring with at most ∆+ 5 colors, where ∆ denotes the maximum degree of the graph. In this paper, we show that ∆ + 4 colors suffice if the graph is planar and without a C3 adjacent to a C4. Moreover, we prove that every planar without C4 and C5 and maximum degree at least 5 admits an incidence coloring with at most ∆+ 3 colors.