Almost all graphs are rigid—revisited

It is shown that almost all graphs are unretractive, i.e. have no endomorphisms other than their automorphisms. A more general result has already been published in [V. Koubek, V. Rödl, On the minimum order of graphs with given semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155]. In the paper at hand, a different proof is presented, following an approach of P. Erdős and A. Rényi that was used in their proof [P. Erdős, A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315] that almost all graphs are asymmetric (have a trivial automorphism group). The approach is modified using an algebraically motivated reduction to idempotent endomorphisms. These take the role of the automorphisms in the proof of Erdős and Rényi. A bound of O(n2(34)n) is provided for the ratio of retractive graphs among all graphs with nn vertices, confirming an earlier statement by Babai [L. Babai, Automorphism groups, isomorphism, reconstruction, in: R.L. Graham, M. Grötschel, L. Lovász (Eds.), in: Handbook of Combinatorics, vol. 2, Elsevier, Amsterdam, 1995, pp. 1447–1540]. The fact that almost all graphs are unretractive and asymmetric can be summarized in the statement that almost all graphs are rigid (have a trivial endomorphism monoid), and the same bound can be obtained for corresponding ratios of nonrigid graphs.