List injective coloring of a class of planar graphs without short cycles

An injective k-coloring of a graph G is a mapping c: V (G) -> {1, 2,..., k} such that c(u) does not satisfy c(v) whenever u, v have a common neighbor in G. A list assignment of a graph G is a mapping L that assigns a color list L(v) to each vertex nu V (G). Given a list assignment L of G, an injective coloring phi of G is called an injective L-coloring if phi(nu) is an element of L(nu) for every nu is an element of V (G). In this paper, we show that if G is a planar graph with girth g >= 5, then chi(l)(i) (G) <= Delta(G) + Delta(G) >= 11.