Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T19:14:46.007Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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