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doi:10.1016/0012-365X(94)90255-0 
Copyright © 1994 Published by Elsevier Science B.V.
The total chromatic number of regular graphs whose complement is bipartite
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J. K. Dugdale
and A. J. W. Hilton*
Received 30 October 1990;
revised 27 March 1992.
Available online 20 January 2003.
Abstract
We show that a regular graph G of order at least 6 whose complement
G" height="15" width="12"> is bipartite has total chromatic number d(G)+1 if and only if1. (i) G is not a complete graph, and
2. (ii)
when n is even.
As an aid in the proof of this, we also show that, for n
4, if the edges of a Hamiltonian path of K2n are coloured with 2n - 1 colours, each edge receiving a different colour, then this can be extended to an edge-colouring of K2n with the same set of 2n–1 colours.
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* Also Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA.
