### Abstract

Given two graphs F and G, an F-WORM coloring of G is an assignment of colors to its vertices in such a way that no F-subgraph of G is monochromatic or rainbow. If G has at least one such coloring, then it is called F-WORM colorable and W^{−}(G,F) denotes the minimum possible number of colors. Here, we consider F-WORM colorings with a fixed 2-connected graph F and prove the following three main results: (1) For every natural number k, there exists a graph G which is F-WORM colorable and W^{−}(G,F)=k; (2) It is NP-complete to decide whether a graph is F-WORM colorable; (3) For each k≥|V(F)|−1, it is NP-complete to decide whether a graph G satisfies W^{−}(G,F)≤k. This remains valid on the class of F-WORM colorable graphs of bounded maximum degree. We also prove that for each n≥3, there exists a graph G and integers r and s such that s≥r+2, G has K_{n}-WORM colorings with exactly r and also with s colors, but it admits no K_{n}-WORM colorings with exactly r+1,…,s−1 colors. Moreover, the difference s−r can be arbitrarily large.

Original language | English |
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Pages (from-to) | 131-138 |

Number of pages | 8 |

Journal | Discrete Applied Mathematics |

Volume | 231 |

DOIs | |

Publication status | Published - Nov 20 2017 |

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### Keywords

- 2-connected graphs
- Feasible set
- Gap in chromatic spectrum
- Lower chromatic number
- WORM coloring

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*231*, 131-138. https://doi.org/10.1016/j.dam.2017.05.008