Summary: Given three planar graphs \(F\), \(H\), and \(G\), an \((F,H)\)-WORM coloring of \(G\) is a vertex coloring such that no subgraph isomorphic to \(F\) is rainbow and no subgraph isomorphic to \(H\) is monochromatic. If \(G\) has at least one \((F,H)\)-WORM coloring, then \(W^{-}_{F,H}(G)\) denotes the minimum number of colors in an \((F,H)\)-WORM coloring of \(G\). We show that

(a)

\(W^{-}_{F,H}(G)\leq 2\) if \(|V(F)|\geq3\) and \(H\) contains a cycle,

(b)

\(W^{-}_{F,H}(G)\leq3\) if \(|V (F)|\geq 4\) and \(H\) is a forest with \(\Delta (H)\geq3\),

(c)

\(W^{-}_{F,H}(G)\leq 4\) if \(|V (F)|\geq 5\) and \(H\) is a forest with \(1\leq\Delta(H)\leq2\).

The cases when both \(F\) and \(H\) are nontrivial paths are more complicated; therefore we consider a relaxation of the original problem. Among others, we prove that any 3-connected plane graph (respectively outerplane graph) admits a 2-coloring such that no facial path on five (respectively four) vertices is monochromatic.