The harmonious chromatic number of complete r-ary trees

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    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 languageEnglish
    Pages (from-to)83-99
    Number of pages17
    JournalDiscrete Mathematics
    Issue number1-3
    Publication statusPublished - 1999


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