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A scheduling problem with a simple graphical solution

Published online by Cambridge University Press:  17 February 2009

A. F. Beecham  and
A. C. Hurley
A. F. Beecham
Affiliation:
CSIRO Division of Chemical Physics, P.O. Box 160, Clayton, Victoria 3168
A. C. Hurley
Affiliation:
CSIRO Division of Chemical Physics, P.O. Box 160, Clayton, Victoria 3168
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Abstract

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It is shown that a problem which arose in the scheduling of two simultaneous competitions between a number of golf clubs may be reduced to that of 4- colouring the edges of a certain bipartite graph which has 4 edges meeting at each vertex. This colouring problem is solved by an analysis in terms of directed cycles, which is simple to carry through in a practical case and is easily extended to the problem with 4 replaced by 2m. The more general colouring problem with 4 replaced by any positive integer is solved by relating it to the marriage problem enunciated by Philip Hall and to the latin multiplication technique of Kaufmann but, in practical applications, this approach involves severe computational difficulties.

Type
Research Article
Information
The ANZIAM Journal , Volume 21 , Issue 4 , April 1980 , pp. 486 - 495
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Hall, P., “On representatives of subsets”, J. Lond. Math. Soc. 10 (1935), 26–30.Google Scholar
[2]Karary, F., Norman, R. Z. and Cartwright, D., Structural models: An introduction to the theory of directed graphs (Wiley, New York, London, Sydney, 1965), p. 386.Google Scholar
[3]Kaufmann, A., Graphs, dynamic programming and finite games (Academic Press, New York, London, 1967), p. 271.Google Scholar
[4]Ore, O., Graphs and their uses (Random House, New York, 1963).Google Scholar