WORM Colorings of Planar Graphs

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 (G) denotes the minimum number of colors in an (F,H)-WORM coloring of G. We show that (a) W (G) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W (G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W (G) ≤ 4 if |V (F)| ≥ 5 and H is a forest with 1 ≤ Δ (H) ≤ 2. 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.