Minimum 2-distance coloring of planar graphs and channel assignment

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. (chi _{2}(G))=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree (Delta ), (chi _2(G) le 7) if (Delta le 3), (chi _2(G) le Delta +5) if (4le Delta le 7) and (chi _2(G) le lfloor frac{3Delta }{2}rfloor +1) if (Delta ge 8). In this paper, we prove that: (1) If G is a planar graph with maximum degree (Delta le 5), then (chi _{2}(G)le 20); (2) If G is a planar graph with maximum degree (Delta ge 6), then (chi _{2}(G)le 5Delta -7).