### Abstract

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. We define Q(m) to be the least positive integer k such that () = m. Then h(G) = Q(m) for any graph G with m edges. We consider the complete r-ary tree of height H, denoted T. We show that for any r = 2, H = 3, if m is the number of edges of T, then h(T) = Q(m), except that h(T) = 7.

Original language | English |
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Pages (from-to) | 83-99 |

Number of pages | 17 |

Journal | Discrete Mathematics |

Volume | 203 |

Issue number | 1-3 |

Publication status | Published - 1999 |

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## Cite this

Edwards, K. (1999). The harmonious chromatic number of complete r-ary trees.

*Discrete Mathematics*,*203*(1-3), 83-99.