### Abstract

Given non-negative integers r, s, and t, an [r,s,t]-coloring of a graph G = (V(G),E(G)) is a mapping c from V(G) ∪ E(G) to the color set {1,...,k} such that {pipe}c(v_{i}) - c(v_{j}){pipe} ≥ r for every two adjacent vertices v_{i},v_{j}, {pipe}c(e_{i}) - c(e_{j}){pipe} ≥ s for every two adjacent edges e_{i},e_{j}, and {pipe}c(v_{i}) - c(e_{j}){pipe} ≥ t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number Χ_{r,s,t}(G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. In this note we examine χ _{1,1,t}(K_{p}) for complete graphs K_{p}. We prove, among others, that χ _{1,1,t}(K_{p}) is equal to p+t-2+min{p,t} whenever t ≥ [p/2]-1, but is strictly larger if p is even and sufficiently large with respect to t. Moreover, as p → ∞ and t=t(p), we asymptotically have χ _{1,1,t}(K_{p})=p+o(p) if and only if t=o(p).

Original language | English |
---|---|

Pages (from-to) | 1041-1050 |

Number of pages | 10 |

Journal | Graphs and Combinatorics |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 1 2013 |

### Fingerprint

### Keywords

- Complete graphs
- Generalized colorings
- [r, s, t]-Chromatic number
- [r, s, t]-Colorings

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*29*(4), 1041-1050. https://doi.org/10.1007/s00373-012-1153-3