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

For a set of nonnegative integers $c_1,\dots ,c_k$, a $(c_1, c_2,\ldots, c_k)$-coloringofagraph$G$isapartitionof$V(G)$into$V_1, \ldots, V_k$suchthatforevery$i$,$1\le i\le k, G[V_i]$hasmaximumdegreeatmost$c_i$. We prove that all planar graphs without 4-cycles and no less than two edges between triangles are $(2,0,0)$-colorable.