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Detachments of Complete Graphs

Published online by Cambridge University Press:  11 April 2005

KEITH EDWARDS
KEITH EDWARDS
Affiliation:
Division of Applied Computing, University of Dundee, Dundee, DD1 4HN, UK (e-mail: kjedwards@dundee.ac.uk)
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Abstract

A detachment of a graph $G$ is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. In this paper we consider the question of whether a graph $H$ is a detachment of some complete graph $K_n$. When $H$ is large and restricted to belong to certain classes of graphs, for example bounded degree planar triangle-free graphs, we obtain necessary and sufficient conditions which give a complete characterization.

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. The results on detachments of complete graphs give exact results on harmonious chromatic number for many classes of graphs, as well as algorithmic results.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 14 , Issue 3 , May 2005 , pp. 275 - 310
Copyright
© 2005 Cambridge University Press

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