Distribution of subgraphs of random regular graphs

The asymptotic distribution of small subgraphs of a random graph has been basically worked out (see Ruciński [5] for example). But for random regular graphs, the main techniques for proving, for instance, asymptotic normality, do not seem to be usable. One very recent result in this direction is to be found in [3], where switchings were applied to cycle counts. The aim of the present note is to show that another very recent method of proving asymptotic normality, given by the authors in [1], can easily be applied to this problem. In particular, it requires considerably less work than using switchings. The application is, however, not direct, in the sense that the result obtained is very weak if the random variable counting copies of a subgraph is examined directly. We obtain a much stronger result by considering isolated copies of a subgraph. To be specific, we investigate the probability space Gn,d of uniformly distributed random d-regular graphs on n vertices (which we assume to be {1, 2, . . . , n}). Asymptotics are for n→∞, and here d is not fixed but may vary with n (though for all our results there is an upper bound on the growth of d, at least implicitly). As usual, we impose the restriction that for the asymptotics, the odd values of n are omitted in the case of odd d. ∗Research supported by NSERC and University of Macau †Research supported by the Canada Research Chairs program and NSERC.