List injective Coloring of Planar Graphs

An injective k-coloring of graph G is a mapping c : V(G) -> {1,2, . . . , k}, such that c(u) not equal c(v) for any two vertices u, v is an element of V(G), whenever u, v have a common neighbor. If G has an injective k-coloring, then we call that G is injective k-colorable. Let chi(i)(G) = min{k vertical bar G is injective k-colorable} be the chromatic number of G. Assign each vertex v is an element of V(G) a coloring set L(v), then L = {L(v)vertical bar v is an element of V(G)} is said to be a color list of G. Let L he a color list of G, if G has an injective coloring c such that c(v) is an element of L(v), for all v is an element of V(G), then we call c is an injective L-coloring of G. If for any color list L, such that vertical bar L(v)vertical bar >= k, G has an injective L-coloring, then G is said to he injective k-choosable. Let chi(l)(i)(G) = min{k vertical bar G is injective k-choosable} be the injective chromatic number of G. In this paper, we prove if G is a planar graph without 3,4,6-cycles and Delta(G) >= 15, then chi(l)(i)(G) <= Delta + 2; If G is a planar graph with g(G) >= 5 and Delta >= 10, then chi(l)(i)(G) <= Delta + 5.