### Abstract

Let G be a simple graph. The achromatic number ?(G) is the largest number of colors possible in a proper vertex coloring of G in which each pair of colors is adjacent somewhere in G. For any positive integer m, let q(m) be the largest integer k such that () = m. We show that the problem of determining the achromatic number of a tree is NP-hard. We further prove that almost all trees T satisfy ?(T) = q(m), where m is the number of edges in T. Lastly, for fixed d and ? > 0, we show that there is an integer N = N(d, ?) such that if G is a graph with maximum degree at most d, and m = N edges, then (1 - ?)q(m) = ?(G) = q(m).

Original language | English |
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Pages (from-to) | 129-136 |

Number of pages | 8 |

Journal | Journal of Graph Theory |

Volume | 26 |

Issue number | 3 |

Publication status | Published - 1997 |

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## Cite this

Cairnie, N., & Edwards, K. (1997). Some Results on the Achromatic Number.

*Journal of Graph Theory*,*26*(3), 129-136.