## Abstract

Let G be an undirected graph with a set of vertices V and a set of edges E. Given an integer r, we assign at most r labels (representing "resources") to each vertex. We say that such an assignment is an r-configuration if, for each label c, the vertices labeled by c form a dominating set for G. In this paper, we are interested in the maximum number, D_{r}(G), of labels that can be assigned to the vertices of graph G by an r-configuration. The decision problem "D_{1}(G) ≥ K?" (known as the domatic number problem) is NP-complete. We first investigate D_{r}(G) for general graphs and establish bounds on D_{r}(G) in terms of D_{1}(G) and the minimum vertex degree of G. We then discuss D_{r}(G) for d-regular graphs. We clearly have D_{r}(G) ≤ r(d + 1). We show that the problem of testing if D_{r}(G) = r(d + 1) is solvable in polynomial time for d-regular graphs with |V| = 2(d + 1), but is NP-complete for those with \V\ = a(d + 1) for some integer a ≥ 3. Finally, we discuss cubic (i.e., 3-regular) graphs. It is easy to show 2r ≤ D_{r}(G) ≤ 4r for cubic graphs. We show that the decision problem for D_{1}(G) = K is co-NP-complete for K = 2 and is NP-complete for K = 4. Although there are many cubic graphs G with D_{1}(G) = 2, surprisingly, every cubic graph has a 2-configuration with five labels, i.e., D_{2}(G) ≥ 5 and such a 2-configuration can be constructed in polynomial time. We use this fact to show D_{r}(G) ≥ ⌊5r/2⌋ in general.

Original language | English |
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Pages (from-to) | 227-254 |

Number of pages | 28 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2000 |

## All Science Journal Classification (ASJC) codes

- General Mathematics