List injective coloring of planar graphs with disjoint 5--cycles

An injective k-coloring of a graph G is called injective if any two vertices joined by a path of length two get different colors. A graph G is injectively k-choosable if for any color list L of admissible colors on V(G) of size k it allows an injective coloring phi such that phi(v) is an element of L(v) whenever v is an element of V(G). Let chi(i)(G), chi(l)(i)(G) denote the injective chromatic number and injective choosability number of (3, respectively. Let G be a plane with disjoint 5(-)-cycles and maximum degree Delta. We show that (1) if Delta >= 12, then chi(l)(i)2(G) <= Delta+ 6; (2) if Delta >= 20, then chi(l)(i)(G) <= ( )Delta + 4.