Injective coloring of planar graphs with girth 5

A k-injective-coloring of a graph G is a mapping c : V (G) -> [1, 2, center dot center dot center dot , k} such that c (u) # c (v) for any two vertices u and v if u and v have a common vertex. The injective chromatic number of G, denoted by chi i (G), is the least k such that G has an injective k-coloring. In this paper, we prove that for planar graph G with g (G) >= 5, Delta (G) >= 20 and without adjacent 5-cycles, chi i (G) <= Delta (G) + 2.