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) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W−F,H(G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W−F,H(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.
@article{bwmeta1.element.doi-10_7151_dmgt_1921,
author = {J. Czap and S. Jendrol' and J. Valiska},
title = {WORM Colorings of Planar Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {353-368},
zbl = {06705133},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1921}
}
J. Czap; S. Jendrol’; J. Valiska. WORM Colorings of Planar Graphs. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 353-368. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1921/