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Counting Coloured Graphs of High Connectivity

Published online by Cambridge University Press:  20 November 2018

Béla Bollobás
Béla Bollobás*
Affiliation:
University of Cambridge, Cambridge, England
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Find exact or asymptotic formulae for the number of labelled graphs of order n having a certain property. The property we are interested in in this note is that of being k-coloured and having connectivity at least l. Special cases of this problem have been tackled by many authors; in particular Gilbert [6], Read [9] and Robinson [11] found exact formulae, and Read and Wright [10], and Wright [12], [13] found asymptotic expressions (for many other examples see [7]). Recently Harary and Robinson [8] counted labelled bipartite blocks, that is 2-connected bipartite graphs. (For terms not defined here and general background in graph theory see [1].) Our present investigations have been prompted by [8]; in particular, as a very special case of our results, we shall prove the conjecture published in [8].

The exact formulae appearing in the enumeration of labelled graphs in general, and in the enumeration of k-coloured labelled graphs in particular, tend to be very pleasing, especially because of the functional equations relating them.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 476 - 484
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bollobâs, B., Extremal graph theory (Academic Press, London, New York and San Francisco, 1978).Google Scholar
2. Bollobâs, B., Graph Theory—An introductory course, Grad. Texts in Math. 63 (Springer-Verlag, New York and Heidelberg, 1979).Google Scholar
3. Bollobâs, B., Degree sequences of random graphs, Discrete Math. 33 (1981), 1–19.Google Scholar
4. Bollobâs, B. and Sauer, N., Uniquely colourable graphs wkth large girth, Can. J. Math. 28 (1976), 1340–1344.Google Scholar
5. Feller, W., An introduction to probability theory and its applications (Wiley and Chapman, New York and London, 1957).Google Scholar
6. Gilbert, E. N., Enumeration of labelled graphs, Can. J. Math. 8 (1956), 405–411.Google Scholar
7. Harary, F. and Palmer, E. M., Graphical enumeration (Academic Press, New York, 1973).Google Scholar
8. Harary, F. and Robinson, W., Labelled bipartite blocks, Can. J. Math. 31 (1979), 60–68.Google Scholar
9. Read, R. C., The number of coloured graphs on labelled nodes, Can. J. Math. 12 (1960), 410–414.Google Scholar
10. Read, R. C. and Wright, E. M., Coloured graphs: A correction and extension, Can. J. Math. 22 (1970), 594–596.Google Scholar
11. Robinson, R. W., Enumeration of non-separable graphs, J. Combinatorial Theory 9 (1970), 327–356.Google Scholar
12. Wright, E. M., Counting coloured graphs, Can. J. Math. 13 (1961), 683–693.Google Scholar
13. Wright, E. M., Counting coloured graphs II, Can. J. Math. 16 (1964), 128–135.Google Scholar