Planar graphs without 4-cycles and close triangles are (2, 0, 0)-colorable

For a set of nonnegative integers $$c_1, \ldots , c_k$$c1,…,ck, a $$(c_1, c_2,\ldots , c_k)$$(c1,c2,…,ck)-coloring of a graph G is a partition of V(G) into $$V_1, \ldots , V_k$$V1,…,Vk such that for every i, $$1\le i\le k, G[V_i]$$1≤i≤k,G[Vi] has maximum degree at most $$c_i$$ci. We prove that all planar graphs without 4-cycles and no less than two edges between triangles are (2, 0, 0)-colorable.