Every Planar Graph Without Adjacent Triangles And 6-cycle Is (3,1,0)-colorable

A graph G is called improper (d1, d2,..., dk)-colorable, or simply (d1, d2,,.., dk)-colorable, if the vertex set of G can be partitioned into subsets V1, V2,..., Vk such that the graph G[Vi] induced by V; has maximum degree at most di, for 1≤i≤k. Let Ω denote the family of planar graphs with neither adjacent triangles nor 6-cycle. In 1976, Steinberg raised the famous conjecture:every planar graph without 4-and 5-cycles is (0, 0,0)-colorable. Based on the conjecture, Borodin and Raspaud conjectured that every planar graph without adjacent triangles and 5-cycle is (0,0,0)-colorable. According to these conjectures, we will prove that every planar graph without adjacent triangles or 6-cycle is (3,1,0)-colorable.