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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

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