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

Abstract

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