The Diameter of Random Graphs

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that if d = d(n) > 3 and m = m(n) satisfy (log n)/d 3 log log n -> oo, 2rf_Imd'/'nd+x log n -» oo and dd~2md~l/nd — log n -» -oo then almost every graph with n labelled vertices and m edges has diameter d. About twenty years ago Erdös [7], [8] used random graphs to tackle problems concerning Ramsey numbers and the relationship between the girth and the chromatic number of a graph. Erdös and Rényi [9], [10] initiated the study of random graphs for their own sake, and proved many beautiful and striking results. The graph invariants investigated in recent years include the clique number [5], [13], [17], the chromatic number [5], [13], the edge chromatic number [11], the circumference [16], [19], and the degree sequence [4]. The aim of this paper is to give rather precise results concerning the diameter. Recall that the diameter diam G of a connected graph is the maximum of the distances between vertices, and a disconnected graph has infinite diameter. The diameter of a random graph has hardly been studied, apart from the case diam G = 2 by Moon and Moser [18], the case diam G < oo by Erdös and Rényi [9], and the diameter of components of sparse graphs by Korshunov [15]. When I was writing this paper, I learned that Klee and Larman [14] proved some results concerning the case diam G = d for fixed values of d. The main result of Klee and Larman [14] is that if d > 3 is a fixed natural number and m = m(n) satisfies md/nd+x log« -» oo and md~l/nd^>0 as n -> oo, then almost every labelled graph with n vertices and m edges has diameter d. As a special case of our results we prove that the conditions above can be weakened to 2d~lmd/nd+l log «^oo and 2d~2md-x/nd log n -* -oo. However, our main aim is to give precise bounds onm = m(n) ensuring that almost every labelled graph with n vertices and m edges has diameter d, where d = d(n) is a function of n which may tend to oo as n -^ oo but which does not increase too fast, say d <\(\ e)log n/log log n. As in our calculations below we are forced to sum estimates d(n) times and d(n) -> oo, we cannot use estimates of the form 0(n~K), o(\), and so on. This is the reason why the paper is so inconveniently full of concrete constants rather than Received by the editors May 12, 1980. 1980 Mathematics Subject Classification. Primary 05C99; Secondary 60C05. © 1981 American Mathematical Society 0002-9947/81/0000-0402/$04.00 41 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use