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Colour Classes for r-Graphs

Published online by Cambridge University Press:  20 November 2018

E. J. Cockayne
E. J. Cockayne*
Affiliation:
University of Victoria, Victoria, B.C.
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By an r-graph G we mean a finite set V(G) of elements called vertices and a set E(G) of some of the r-subsets of V(G) called edges. This paper defines certain colour classes of r-graphs which connect the material of a variety of recent graph theoretic literature in that many existing results may be reformulated as structural properties of the classes for some special cases of r-graphs. It is shown that the concepts of Ramsey Numbers, chromatic number and index may be defined in terms of these classes. These concepts and some of their properties are generalized. The final subsection compares two existing bounds for the chromatic number of a graph.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 3 , September 1972 , pp. 349 - 354
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Ryser, H. J., Combinatorial mathematics, Carus Math. Monograph, Math. Assoc. America, 14 (1965), 38-43.Google Scholar
2. Ramsey, F. P., On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286.Google Scholar
3. Cockayne, E. J., An application of Ramsey's theorem, Canad. Math. Bull. (1) 13 (1970), 145-6.Google Scholar
4. Greenwood, R. E. and Gleason, A. M., Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955), 1-7.Google Scholar
5. Chartrand, G., Geller, D. and Hedetniemi, S., A generalization of the chromatic number, Proc. Cambridge Philos. Soc. 64 (1968), 265-271.Google Scholar
6. Hedetniemi, S., Disconnected colourings of graphs, Combinatorial Structures and their applications. Gordon & Breach, New York (1970), 163-168.Google Scholar
7. G. Szekeres and Wilf, H. S., An inequality for the chromatic number of a graph, J. Comb. Theory (1) 4 (1968), 1-3.Google Scholar
8. Harary, F., Graph theory, Addison-Wesley, Reading, Mass., 1969.Google Scholar