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The Harmonious Chromatic Number of Almost All Trees

Published online by Cambridge University Press:  12 September 2008

Keith Edwards
Keith Edwards
Affiliation:
Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, U.K.
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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.

For any positive integer m, let Q(m) be the least positive integer k such that ≥ m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 1 , March 1995 , pp. 31 - 46
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Edwards, K. J. and McDiarmid, C. J. H. (1994) New Upper Bounds on Harmonious Colorings. J. Graph Theory. 18 257–267.Google Scholar
[2]Edwards, K. J. and McDiarmid, C. J. H. (to appear) The Complexity of Harmonious Colouring for Trees. Disc. Appl. Maths..Google Scholar
[3]Frank, O., Harary, F. and Plantholt, M. (1982) The line-distinguishing chromatic number of a graph. Ars Combinatoria 14 241–252.Google Scholar
[4]Goh, W. M. Y. and Schmutz, E. (1994) Unlabelled Trees: Distribution of the Maximum Degree. Random Struct. Alg. 5 411–440.CrossRefGoogle Scholar
[5]Harary, F. and Palmer, E. M. (1973) Graphical Enumeration. Academic Press.Google Scholar
[6]Hopcroft, J. and Krishnamoorthy, M.S. (1983) On the harmonious coloring of graphs. SIAM J. Alg. Disc. Methods 4 306–311.CrossRefGoogle Scholar
[7]Lu, Z. (1993) The harmonious chromatic number of a complete binary and trinary tree. Disc. Maths. 118 165–172.Google Scholar
[8]McDiarmid, C. J. H. Personal communication.Google Scholar
[9]Miller, Z. and Pritikin, D. (1991) The harmonious colouring number of a graph. Disc. Maths. 93 211–228.CrossRefGoogle Scholar
[10]Mitchem, J. (1989) On the harmonious chromatic number of a graph. Disc. Maths. 74 151–157.CrossRefGoogle Scholar
[11]Otter, R. (1948) The number of trees. Ann. Math. 49 583–599.Google Scholar
[12]Robinson, R. W. and Schwenk, A. J. (1975) The Distribution of Degrees in a Large Random Tree. Disc. Maths. 12 359–372.CrossRefGoogle Scholar
[13]Schwenk, A. J. (1977) An Asymptotic Evaluation of the Cycle Index of a Symmetric Group. Disc. Maths. 18 71–78.CrossRefGoogle Scholar
[14]Wilson, B. (1990) Line distinguishing and harmonious colourings. In: Nelson, R. and Wilson, R. J. (eds.) Graph Colourings, Pitman Research Notes in Mathematics 218 115–133, Longman Scientific & Technical.Google Scholar