### Abstract

For an integer k≥1, the k-improper upper chromatic number χ̄_{k-imp}(G) of a graph G is introduced here as the maximum number of colors permitted to color the vertices of G such that, for any vertex ν in G, at most k vertices in the neighborhood N(ν) of ν receive colors different from that of ν. The exact value of χ̄_{k-imp}is determined for several types of graphs, and general estimates are given in terms of various graph invariants, e.g. minimum and maximum degree, vertex covering number, domination number and neighborhood number. Along with bounds on χ̄_{k-imp} for Cartesian products of graphs, exact results are found for hypercubes. Also, the analogue of the NordhausGaddum theorem is proved. Moreover, the algorithmic complexity of determining χ̄ _{k-imp} is studied, and structural correspondence between k-improper C-colorings and certain kinds of edge cuts is shown.

Original language | English |
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Pages (from-to) | 174-186 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 159 |

Issue number | 4 |

DOIs | |

Publication status | Published - Feb 28 2011 |

### Keywords

- 3-consecutive coloring
- Edge cut
- Graph improper coloring
- Upper chromatic number
- k-improper C-coloring

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*159*(4), 174-186. https://doi.org/10.1016/j.dam.2010.11.004