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On the Non-Existence of a Type of Regular Graphs of Girth 5

Published online by Cambridge University Press:  20 November 2018

William G. Brown
William G. Brown*
Affiliation:
University of British Columbia
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ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that

1.1

Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 644 - 648
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Berge, C., Théorie des graphes et ses applications (Paris, 1958).Google Scholar
2. Brown, W. G., On Hamiltonian regular graphs of girth six, J. London Math. Soc., to appear.Google Scholar
3. Erdös, P. and Sachs, H., Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Univ. Halle, Math.-Nat., 12 (1963), 251–258.Google Scholar
4. Hoffman, A. J. and Singleton, R. R., On Moore graphs with diameters 2 and 3, IBM J. Res. Develop., 4 (1960), 497–504.Google Scholar
5. Mirsky, L., An introduction to linear algebra (Oxford, 1955).Google Scholar
6. Robertson, N., The smallest graph of girth 5 and valency 4, Bull. Amer. Math. Soc., 70 (1964), 824–825.Google Scholar