## Abstract

Let G be a simple graph with vertex set V (G) and edge set E(G). A subset S of V (G) is called an independent set if no two vertices of S are adjacent in G. The minimum number of independent sets which form a partition of V (G) is called chromatic number of G, denoted by χ(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum number of edge covers which form a partition of E(G) is called edge covering chromatic number of G, denoted by χ′_{c}(G). Given nonnegative integers r, s, t and c, an [r, s, c, t]-coloring of G is a mapping f from V (G)∪(G) to the color set {0, 1,..., k-1} such that the vertices with the same color form an independent set of G, the edges with the same color form an edge cover of G, and {pipe}f(v_{i})-f(v_{j}){pipe} ≥ r if v_{i} and v_{j} are adjacent, {pipe}f(e_{i})-f(e_{j}){pipe} ≥ s for every e_{i}, e_{j} from different edge covers, {pipe}f(v_{i}) - f(e_{j}){pipe} ≥ t for all pairs of incident vertices and edges, respectively, and the number of edge covers formed by the coloring of edges is exactly c. The [r, s, c, t]-chromatic number χ_{r;s;c;t}(G) of G is deffined to be the minimum k such that G admits an [r, s, c, t]-coloring. In this paper, we present the exact value of χ_{r,s,c,t}(G) when δ(G) = 1 or G is an even cycle.

Original language | English (US) |
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Pages (from-to) | 165-169 |

Number of pages | 5 |

Journal | Bulletin of the Malaysian Mathematical Sciences Society |

Volume | 34 |

Issue number | 1 |

State | Published - 2011 |

## Keywords

- Chromatic number
- Edge covering coloring
- [r, s, c, t]-coloring
- [r, s, t]-coloring

## ASJC Scopus subject areas

- Mathematics(all)